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1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −0.1 (which equals the negative value of the reciprocal of 10). |
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Both 10±1 are primes, both (10±2)/2 are primes, both (10±3)/3 are primes, both 10±4 are powers of 2 (also common [[radix|base]]s for computer, base 8 and base 14), both 10±5 (also 10±7 and 10±E) are primes, and both 10±6 are regular numbers (i.e. 3-smooth numbers) (i.e. numbers ''n'' such that 1/''n'' terminates in dozenal) (also twice a power of 3). |
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10 (twelve, dozen, onety) is the [[radix|base]] of the [[numeral system]] of this wiki. |
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Both 100±1 are semiprimes, both (100±2)/2 are primes, (100−''d'')/''d'' is prime for all divisors ''d'' of 10 except 1 and 4 (note that this number for ''d'' = 1 and 4 cannot be primes, because of algebra factors: 100−1 = (10−1) × (10+1), and 100−4 = (10−2) × (10+2), and can only be semiprimes), both 100±5 are primes, both 100±7 are primes, both (100±8)/8 are primes, both 100±E are semiprimes, both (100±10)/10 are primes (Note: (100−3)/3 and (100−6)/6 are primes, but (100+3)/3 and (100+6)/6 are prime squares, also, (100+9)/9 is prime, but (100−9)/9 also has algebra factors (100−9 = (10−3) × (10+3)) and cannot be prime) |
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An interesting thing is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −1/10 = −0.1, i.e. |
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All [[square number|squares]] end with square digits (i.e. end with 0, 1, 4 or 9), if ''n'' is divisible by both 2 and 3, then ''n''<sup>2</sup> ends with 0, if ''n'' is not divisible by 2 or 3, then ''n''<sup>2</sup> ends with 1, if ''n'' is divisible by 2 but not by 3, then ''n''<sup>2</sup> ends with 4, if ''n'' is not divisible by 2 but by 3, then ''n''<sup>2</sup> ends with 9. If the unit digit of ''n''<sup>2</sup> is 0, then the dozens digit of ''n''<sup>2</sup> is either 0 or 3, if the unit digit of ''n''<sup>2</sup> is 1, then the dozens digit of ''n''<sup>2</sup> is even, if the unit digit of ''n''<sup>2</sup> is 4, then the dozen digit of ''n''<sup>2</sup> is 0, 1, 4, 5, 8 or 9, if the unit digit of ''n''<sup>2</sup> is 9, then the dozen digit of ''n''<sup>2</sup> is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1) |
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<math>\sum_{k=1}^{\infty}(k)=-\frac{1}{10}=-0.1</math> |
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(this number is |
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The numbers ''n'' such that the concatenation of ''n'' and the unit (1), i.e. 10''n''+1 (all squares of primes except 4 and 9 are of this form), is square, are all [[even number]]s, and the half of these ''n'' are exactly the [[generalized pentagonal number]]s, and such numbers are important to [[Euler]]'s theory of [[Partition (number theory)|partition]]s, as expressed in his [[pentagonal number theorem]] (the '''pentagonal number theorem''', originally due to [[Leonhard Euler|Euler]], relates the product and series representations of the [[Euler function]]. It states that |
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<math>\zeta(-1)</math> |
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, which equals the negative value of the reciprocal of 10, or the additive inverse of the multiplicative inverse of 10) i.e. |
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:<math>\prod_{n=1}^{\infty}\left(1-x^{n}\right)=\sum_{k=-\infty}^{\infty}\left(-1\right)^{k}x^{k\left(3k-1\right)/2}=1+\sum_{k=1}^\infty(-1)^k\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\right).</math> |
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<math>\zeta(-1)=-\frac{1}{10}=-0.1</math> |
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In other words, |
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The negative value of the reciprocal of 10 (i.e. −1/10) is the value counter-intuitively ascribed to the series 1+2+3.... |
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:<math>(1-x)(1-x^2)(1-x^3)(1-x^4) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{10} - x^{13} + x^{1\mathcal{X}} + x^{22} - \cdots.</math> |
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Both 10±1 are primes, both 10±2 are even semiprimes, both 10±3 are odd semiprimes, both 10±4 are powers of 2, both 10±5 (also 10±7 and 10±E) are primes, and both 10±6 are regular numbers (i.e. 3-smooth numbers) (i.e. numbers n such that 1/n terminates in dozenal). |
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The exponents 1, 2, 5, 7, 10, 13, 1X, 22, ... on the right hand side are given by the formula {{math|''g''<sub>''k''</sub> {{=}} ''k''(3''k'' − 1)/2}} for ''k'' = 1, −1, 2, −2, 3, −3, 4, −4, ... and are called (generalized) [[pentagonal numbers]]. |
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Both radix 10±4 are used in computing as a more compact representation of binary (1 base-8 digit per 3 bits, and 1 base-14 digit per 4 bits). |
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This holds as an identity of convergent [[power series]] for <math>|x|<1</math>, and also as an identity of [[formal power series]]. |
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A striking feature of this formula is the amount of cancellation in the expansion of the product), also, the identity implies a marvelous [[Recurrence relation|recurrence]] for calculating <math>p(n)</math>, the number of [[partition (number theory)|partitions]] of ''n'' (''p''(''n'')): |
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1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −0.1 (which equals the negative value of the reciprocal of 10) |
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:<math>p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-10)+p(n-13)-p(n-1\mathcal{X})-p(n-22)+\cdots</math> |
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10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + X0 + E0 + 10 + ... = −1 (the sum of all positive multiples of 10 is −1) |
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:<math>p(0)=1</math> |
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10 is a [[composite number]], the smallest positive integer with exactly six [[divisor]]s, its divisors being 1, 2, 3, 4, 6 and 10. 10 is also a [[highly composite number]], the next one being 20 (=2×10), and all highly composite numbers ≥10 are divisible by 10. |
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or more formally, |
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10 is the smallest [[radix|base]] such that all of the five simplest fractions (1/2, 1/3, 2/3, 1/4, and 3/4) all only have one digit (0.6, 0.4, 0.8, 0.3, and 0.9). |
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:<math>p(n)=\sum_k (-1)^{k-1}p(n-g_k)</math> |
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10 is the smallest abundant number. |
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Also the [[Divisor function|sum of divisors]] of ''n'' (σ(''n'')): |
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10 is the average of a twin prime pair (E and 11), i.e. both 10−1 and 10+1 are primes. |
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:<math>\sigma(n)=\sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+\sigma(n-10)+\sigma(n-13)-\sigma(n-1\mathcal{X})-\sigma(n-22)+\cdots</math> |
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10 is the least common multiple of the n such that a<sub>n</sub>x<sup>n</sup>+a<sub>n-1</sub>x<sup>n-1</sup>+...+a<sub>1</sub>x+a<sub>0</sub>=0 have algebraic solution: {1, 2, 3, 4}. |
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but if the last term is σ(0) (this situation appears if and only if ''n'' itself is [[generalized pentagonal number]], i.e. the concatenation of 2''n'' and 1 is square), then we change it to ''n''. |
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10 is the least common multiple of the n such that regular n-gon (regular polygon with n sides) can tile the plane: {3, 4, 6}. |
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where the summation is over all nonzero integers ''k'' (positive and negative) and <math>g_k </math> is the ''k''<sup>th</sup> generalized pentagonal number. Since <math>p(n)=0</math> for all <math>n<0</math>, the series will eventually become zeroes, enabling discrete calculation, besides, generalized pentagonal numbers are closely related to [[centered hexagonal number]]s (also called [[hex number]]s, the hex numbers are end with 1, 7, 7, 1, 1, 7, 7, 1, ...). When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper. |
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10 is the least common multiple of the all-harshad numbers: {1, 2, 4, 6}. (10 itself is a harshad number in all bases except base 8 (octal)) |
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The [[digital root]] of a square is 1, 3, 4, 5 or E. |
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The numbers n such that all primes not dividing n are = ±1 mod n (i.e. either of the form nk+1 or of the form nk−1) are exactly the proper divisors of 10: {1, 2, 3, 4, 6}, and the least common multiple of these numbers are exactly 10. |
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No [[repdigit]]s with more than one digit are squares, in fact, a square cannot end with three same digits except 000. |
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The numbers n such that cos(1/n turns) is rational (in fact, 2cos(1/n turns) is integer) are exactly the proper divisors of 10: {1, 2, 3, 4, 6}, and the least common multiple of these numbers are exactly 10. |
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No four-digit [[palindromic number]]s are squares. (we can easily to prove it, since all four-digit palindromic number are divisible by 11, and since they are squares, thus they must be divisible by 11<sup>2</sup> = 121, and the only four-digit palindromic number divisible by 121 are 1331, 2662, 3993, 5225, 6556, 7887, 8EE8, 9119, X44X and E77E, but none of them are squares) |
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10 is the first number n such that both n and n! can be written as product of >1 factorials of primes: 10 = 2!*3! and 10! = 2!*3!*E! |
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{|class="wikitable" |
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10 is the only number which can be written as x^y+x^(y+1) with x≥2, y≥1 in 2 different ways. |
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!''n'' |
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!''n''-digit palindromic squares |
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!square roots |
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!number of ''n''-digit palindromic squares |
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|- |
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!1 |
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|1, 4, 9||1, 2, 3||3 |
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|- |
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!2 |
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|none||none||0 |
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|- |
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!3 |
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|121, 484||11, 22||2 |
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|- |
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!4 |
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|none||none||0 |
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|- |
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!5 |
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|10201, 12321, 14641, 16661, 16E61, 40804, 41414, 44944||101, 111, 121, 12E, 131, 202, 204, 212||8 |
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|- |
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!6 |
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|160061||42E||1 |
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|- |
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!7 |
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|1002001, 102X201, 1093901, 1234321, 148X841, 4008004, 445X544, 49XXX94||1001, 1015, 1047, 1111, 1221, 2002, 2112, 2244||8 |
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|- |
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!8 |
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|none||none||0 |
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|- |
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!9 |
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|100020001, 102030201, 104060401, 1060E0601, 121242121, 123454321, 125686521, 1420E0241, 1444X4441, 1468E8641, 14X797X41, 1621E1261, 163151361, 1XX222XX1, 400080004, 404090404, 410212014, 4414X4144, 4456E6544, 496787694, 963848369||10001, 10101, 10201, 10301, 11011, 11111, 11211, 11E21, 12021, 12121, 1229E, 1292E, 12977, 14685, 20002, 20102, 20304, 21012, 21112, 22344, 31053||19 |
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|- |
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!X |
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|1642662461||434X5||1 |
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|- |
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!E |
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|10000200001, 10221412201, 10444X44401, 12102420121, 12345654321, 141E1E1E141, 14404X40441, 16497679461, 40000800004, 40441X14404, 41496869414, 44104X40144, 49635653694||100001, 101101, 102201, 110011, 111111, 11E13E, 120021, 12X391, 200002, 201102, 204204, 210012, 223344||11 |
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|- |
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!10 |
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|none||none||0 |
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|} |
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It is conjectured that if ''n'' is divisible by 4, then there are no ''n''-digit palindromic squares. |
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10 is the ratio of 20 and 2, where 20 is the largest n such that lambda(n)=2, and 2 is the largest n such that lambda(n)=1, where lambda is the Carmichael lambda function. |
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''R''<sub>''n''</sub><sup>2</sup> (where ''R''<sub>''n''</sub> is the [[repunit]] with length ''n'') is a palindromic number for ''n'' ≤ E, but not for ''n'' ≥ 10 (thus, for all odd number ''n'' ≤ 19, there is ''n''-digit palindromic square 123...321), besides, 11<sup>''n''</sup> (also 1{0}1<sup>''n''</sup>, i.e. 101<sup>''n''</sup>, 1001<sup>''n''</sup>, 10001<sup>''n''</sup>, etc.) is a palindromic number for ''n'' ≤ 5, but not for ''n'' ≥ 6, and it is conjectured that no palindromic numbers are ''n''-th powers if ''n'' ≥ 6. |
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10 is the [[least common multiple]] of 1 to [[4]]. |
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The square numbers using no more than two distinct digits are 0, 1, 4, 9, 14, 21, 30, 41, 54, 69, 84, X1, 100, 121, 144, 344, 400, 441, 484, 554, 900, 3000, 4344, 9944, 10000, 11XX1, 16661, 40000, 41414, 44944, 47744, 66969, 90000, 111101, 114144, 300000, 444404, 454554, 999909, 1000000, 1141144, 3333030, 4000000, 4544554, 9000000, 11110100, 30000000, 41144144, 44440400, 99990900, XXXXXXX1, 100000000, 333303000, 400000000, 900000000, 1111010000, 3000000000, 4444040000, 9999090000, 10000000000, 33330300000, 40000000000, 90000000000, 111101000000, 300000000000, 444404000000, 999909000000, 1000000000000, ... |
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10 is the [[greatest common divisor]] of the order of unsolvable groups ≤ 14000. |
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41414 is the largest undulating square (of the form ''ababab...''), note that the only 2 distinct digits in it (1 and 4) and the only 2 distinct two-digit numbers formed by its digits (14 and 41) are also all squares. |
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10 is the [[greatest common divisor]] of the [[composite number|composite]] order of simple groups ≤ 14000. |
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A [[cube number|cube]] can end with all digits except 2, 6 and X (in fact, no [[perfect power]]s end with 2, 6 or X), if ''n'' is not congruent to 2 mod 4, then ''n''<sup>3</sup> ends with the same digit as ''n''; if ''n'' is congruent to 2 mod 4, then ''n''<sup>3</sup> ends with the digit (the last digit of ''n'' +− 6). |
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10 is the product of the n such that there are no Greco-Latin squares of order n: {2, 6}. |
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The cube numbers using no more than two distinct digits are 0, 1, 8, 23, 54, X5, 1000, 1331, 8000, 1000000, 8000000, 1000000000, 8000000000, 1000000000000, ... |
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The [[star number]]s are exactly the [[centered polygonal number|centered n-gonal numbers]] for n=10. |
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The [[digital root]] of a cube can be any number. |
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10 is the value of |
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21 and 201 are both squares, and it is conjectured that no other numbers of the form 2000...0001 (i.e. of the form 2×10<sup>''n''</sup>+1) are squares. |
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<math>\frac{\pi^2}{(\ln 2) \cdot (\ln \lambda)}</math> |
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, where |
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10814 and 100814 are both squares, and 10854 and 100853 are both cubes. |
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<math>\lambda</math> |
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is the [[Levy’s constant]]. 10 is the smallest nonnegative even number ''n'' such that the numerator of |
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If ''k''≥2, then ''n''<sup>''k''+2</sup> ends with the same digit as ''n''<sup>''k''</sup>, thus, if ''i''≥2, ''j''≥2 and ''i'' and ''j'' have the same [[Parity (mathematics)|parity]], then ''n''<sup>''i''</sup> and ''n''<sup>''j''</sup> end with the same digit. |
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<math>\frac{\zeta(n)}{\pi^n}</math> |
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is not 1. |
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If ''x''^2 + ''y''^2 = ''z''^2 (that is, {''x'', ''y'', ''z''} is a [[Pythagorean triple]], then ''xy'' end with 0 (and thus ''xyz'' also end with 0). |
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10 is the smallest nonnegative even number ''n'' such that the numerator of ''B<sub>n</sub>'' has at least one prime factor ≥ ''n''+3, where ''B<sub>n</sub>'' is the [[Bernoulli number]]. |
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Squares (and every powers) of 0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, E3854, 1E3854, X08369, ... end with the same digits as the number itself. (since they are [[automorphic number]]s, from the only four solutions of ''x''<sup>2</sup>−''x''=0 in the [[Ring (mathematics)|ring]] of [[p-adic number|10-adic number]]s (dozadic numbers), these solutions are 0, 1, ...2E21E61E3854 and ...909X05X08369, since 10 is neither a prime nor a [[prime power]], the ring of the 10-adic numbers is not a [[Field (mathematics)|field]], thus there are solutions other than 0 and 1 for this equation in 10-adic numbers) |
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It is known that |
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The triangular numbers using no more than two distinct digits are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, 77, 89, X0, E4, 191, 303, 446, 550, 633, 66X, 6X6, 1117, 3X3X, 3EE3, 6060, 6161, 6366, 6999, 6EE6, 8989, 9779, 23223, 35553, 50050, 77677, 113113, 303333, 331331, 600600, X33X33, 3030330, 60006000, 333666333, 6000060000, 600000600000, ... |
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<math>\zeta(n)</math> |
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is irrational for ''n''=3, and for at least one other odd ''n''≤10. |
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The pronic numbers using no more than two distinct digits are 0, 2, 6, 10, 18, 26, 36, 48, 60, 76, 92, E0, 110, 606, 656, 992, XX0, EE6, 1118, 2232, 7878, EE00, 10100, 33330, 46446, 6XXX6, X00X0, 118118, 226226, 606666, 662662, EEE000, 1001000, 6060660, EEEE0000, 100010000, EEEEE00000, 10000100000, EEEEEE000000, 1000001000000, ... |
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A number ''n'' has terminate reciprocal if and only if ''n'' is regular to 10, i.e. all prime divisors of ''n'' are prime divisors of 10, besides, a number ''n'' has reciprocal with only 1 digit after the dozenal point if and only if ''n'' is divisor of 10. |
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Except for 6 and 24, all even [[perfect number]]s end with 54. Additionally, except for 6, 24 and 354, all even perfect numbers end with 054 or 854. Besides, if any odd perfect number exists, then it must end with 1, 09, 39, 69 or 99. |
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The ''n''-th Fibonacci number ''F''(''n'') has terminate reciprocal if and only if ''n'' is divisor of 10. |
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The [[digital root]] of an even perfect number is 1, 4, 6 or X. |
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10 is the smallest [[abundant number]], since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 14) is greater than itself. |
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Since 10 is the smallest [[abundant number]], all numbers end with 0 are abundant numbers, besides, all numbers end with 6 except 6 itself are also abundant numbers. |
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10 is the smallest positive integer which is not [[power]] of [[squarefree number]] (i.e. not in [https://oeis.org/A072774 OEIS A072774]). |
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10 is the smallest positive integer which is neither [[squarefree]] nor [[prime power]]. |
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10 is the smallest positive integer which is neither [[squarefree]] nor [[perfect power]]. |
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10 is the smallest positive integer which is neither [[squarefree]] nor [[powerful]]. |
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10 is the smallest number n>1 such that the n-th cyclotomic polynomial has no root mod p for all primes p <= n. |
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10 is the smallest number k such that k/(largest power of squarefree kernel of k) is larger than 1, i.e. 10/6 = 2 > 1. |
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10 is the smallest number n such that (n * Product_{primes p|n} p) ([https://oeis.org/A064549 OEIS A064549]) is not perfect power (note that (n * Product_{primes p|n} p) is powerful for all n), and for n=10, (n * Product_{primes p|n} p) is exactly the smallest Achilles number (powerful but not perfect power) 60 |
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The [[highly composite number]]s contains the first 5 [[multiple]]s of 10 (10, [[20]], [[30]], [[40]], and [[50]]), and no other numbers between them. |
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10 is the only number n such that for all prime factors p of n, p^(largest r such that p^r divides n) + p^(smallest r such that p^r does not divide n) = n. (i.e. for p=2, 2^2 + 2^3 = 10, and for p=3, 3^1 + 3^2 = 10) |
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10 is the only number which is both (equal to ''mn'') and (average of 2<sup>''m''</sup> and 2<sup>''n''</sup>) for some integer pairs (''m'', ''n'') such that ''m'' and ''n'' are consecutive integers (i.e. ''n'' = ''m''+1). ((''m'', ''n'') = (3, 4)) |
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10 is the largest number n such that if 0≤k≤n−1 and gcd(k, n) = 1 (i.e. k and n are coprime), then n+k is prime. |
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Both of 10−1 and 10+1 are primes, in fact, all of 10±1, (10/2)±1, and (10/3)±1 are primes, the next such number is 220990, besides, all such numbers must be divisible by 10. |
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10 is the sum of the numbers in the first [[Pythagorean triple]]: {3, 4, 5}. |
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10−2 is the smallest noncototient, 10+2 is the smallest nontotient, the average of them is 10. |
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10−2 is the length of the longest cycle in “sum of square of digits”, 10+2 is the length of the longest cycle in “sum of cube of digits”, the average of them is 10. (note that the smallest number in these two cycles are both 8) |
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10 is the average of the smallest nontotient (12) and the smallest noncototient (X). |
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10 is the first [[sublime number]], a number that has a perfect number (6) of divisors, and the sum of its divisors is also a perfect number (24), there are only two known such numbers (including 10 itself). |
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10 is the product of the first three [[factorial]]s (not including 0! = 1). |
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10 is the smallest [[radix|base]] b such that the number of trailing zeros in n! (the [[factorial]] of n) does not depend on one of the primes or prime powers dividing b. |
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The square of any prime >3 is congruent to 1 mod 10 (in fact, congruent to 1 mod 20). |
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The sum of any pair of [[twin prime]]s (other than 3 and 5) is divisible by 10. (i.e. end with 0) |
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The product of any pair of [[twin prime]]s (other than 3 and 5) is congruent to E mod 10. (i.e. end with E) |
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10 is the largest number which is a divisor of the sum of all pairs of twin primes other than (3, 5). |
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10 is the largest number which is a divisor of the sum of all but finitely many pairs of twin primes (assuming that there are infinitely many pairs of twin primes). |
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10 is the largest number which is a divisor of xy for all Pythagorean triples {x, y, z} (x≤y≤z). |
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10 is the largest [[radix|base]] such that both "all squares end with square digits" and "all primes not dividing the base end with either prime digits not dividing the base or 1" are true. |
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10 is the largest base such that all these three properties are true: |
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{|class="wikitable" |
{|class="wikitable" |
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|+ unit digit of ''n''<sup>''k'' |
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|Property of the numerical system |
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!{{diagonal split header|''n''|''k''}}!!0!!1!!2!!3!!4!!5!!6!!7!!8!!9!!X!!E!!10!!11!!12!!13!!14!!15!!16!!17!!18!!19!!1X!!1E!!20!!Period |
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|Property of the base (b) |
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|Bases (Note: 1 cannot be the base of the system) |
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|- |
|- |
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!0 |
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|All squares end with square digits |
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|1||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
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|[https://oeis.org/A254328 Numbers n such that all x^2 mod n are squares] |
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!1 |
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|1, 2, 3, 4, 5, 8, 10, 14 |
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|- |
|- |
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!1 |
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|All primes not dividing the base end with either prime digits not dividing the base or 1 (unit) |
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|1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1 |
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|[https://oeis.org/A048597 Numbers n such that reduced residue system of n consists of only primes and 1] |
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!1 |
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|1, 2, 3, 4, 6, 8, 10, 16, 20, 26 |
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|- |
|- |
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!2 |
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|All squares of the primes not dividing the base end with 1 |
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|1||2||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4 |
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|[https://oeis.org/A018253 Numbers n with the property m^2 == 1 (mod n) for all integer m coprime to n] |
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!2 |
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|1, 2, 3, 4, 6, 8, 10, 20 (exactly the divisors of 20) |
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|- |
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!3 |
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|1||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9 |
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!2 |
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|- |
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!4 |
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|1||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4 |
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!1 |
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|- |
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!5 |
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|1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1 |
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!2 |
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|- |
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!6 |
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|1||6||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
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!1 |
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|- |
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!7 |
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|1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1 |
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!2 |
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|- |
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!8 |
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|1||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4 |
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!2 |
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|- |
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!9 |
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|1||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9 |
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!1 |
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|- |
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!X |
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|1||X||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4 |
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!1 |
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|- |
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!E |
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|1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1 |
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!2 |
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|- |
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!10 |
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|1||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
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!1 |
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|- |
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!11 |
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|1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1 |
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!1 |
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|- |
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!12 |
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|1||2||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4 |
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!2 |
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|- |
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!13 |
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|1||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9||3||9 |
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!2 |
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|- |
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!14 |
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|1||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4 |
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!1 |
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|- |
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!15 |
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|1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1||5||1 |
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!2 |
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|- |
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!16 |
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|1||6||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
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!1 |
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|- |
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!17 |
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|1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1||7||1 |
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!2 |
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|- |
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!18 |
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|1||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4||8||4 |
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!2 |
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|- |
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!19 |
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|1||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9||9 |
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!1 |
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|- |
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!1X |
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|1||X||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4||4 |
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!1 |
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|- |
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!1E |
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|1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1||E||1 |
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!2 |
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|- |
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!20 |
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|1||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
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!1 |
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|} |
|} |
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The period of the unit digits of powers of a number must be a divisor of 2 (= λ(10), where λ is the [[Carmichael function]]). |
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10 is the largest [[radix|base]] such that both "all squares end with square digits" and "all primes end with prime digits or 1" are true. |
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{|class="wikitable" |
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10 is the largest number n with the property that every number m in the range n < m < 2n that is coprime to n is also prime. |
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!''n'' |
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!possible unit digit of an ''n''th power |
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10 is the largest number n with the property that every number m in the range 1 < m ≤ 2n that is coprime to n is also prime. |
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|- |
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|0 |
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10 is the largest natural number ''n'' such that for all even numbers ''k'' dividing ''n'', both ''k''−1 and ''k''+1 are noncomposite (including 0). |
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|1 |
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|- |
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10 is the largest natural number ''n'' such that for all numbers ''k'' dividing ''n'', both ''k''−1 and ''k''+1 are either prime or power of 2 (or 0). |
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|1 |
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|any number |
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10 is the largest ''n'' such that all even divisors >2 of ''n'' are average of a twin prime pair. |
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|- |
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|even number ≥ 2 |
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10 is the largest number k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1. |
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|0, 1, 4, 9 (the square digits) |
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|- |
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10 is the only number ''n'' ≤ 220000 such that ''n''±1, ''n''/2±1 and ''n''/3±1 are all primes. (the next such ''n'' is 220990, note that this number contains 3 distinct digits all twice) (in fact, all such ''n'' except 10 is divisible by 890) |
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|odd number ≥ 3 |
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|0, 1, 3, 4, 5, 7, 8, 9, E (all digits != 2 mod 4) |
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10 is conjectured to be the largest number n such that there is no x≤n such that both n*x + n + x and n*x − n − x are primes. (for n=10, the smallest such x is exactly the smallest number > 10 (i.e. 11)) |
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|} |
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10!+1 is divisible by 121 = 11^2, and 10 is the only known non-semiprime n such that n!+1 is divisible by (n+1)^2. |
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10 is the smallest integer b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p. (the other such b≤100 are 39, 68, 76, and 100) (Given b, the number of trailing zeros at the end of the base-b representation of x! is asymptotic to x/M where M is the maximum over p|b of (p-1)*v_p(b). Usually only one prime p attains the maximum and then the number is v_p(x!)/v_p(b) for all but finitely many x. But for b = 10, 39, 68, 76, 100, ..., at least two v_p(x!) must be computed) |
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10 is the smallest [[radix|base]] n for which there exists k≤n (in fact, k≤n<sup>2</sup>) such that k! (where ! is the [[factorial]]) is not [[harshad number]] base n (k=7). |
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The [[Euclid number]] ''n''#+1 (where # is the primorial) is prime for all ''n''≤10 (but not for ''n''=11). |
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14060 is the smallest number divisible by all natural numbers from 1 to 10, it is also the largest positive integer n such that for any m < n, and for all values of k, d(n^k) > d(m^k). (Let k and m represent positive integers only) |
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Regular 10-gon is the highest polygon in the [[tessellation]]. |
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The numbers ''n'' ≤ 10 with terminate reciprocal are {1, 2, 3, 4, 6, 8, 9, 10}, and the divisors of 10 are {1, 2, 3, 4, 6, 10}, the difference of these two sets are 8 and 9 (of course, the second set is a subset of the first set), (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the [[multiplicative inverse|reciprocal]] of all these ''n'' except 8 and 9 are all 1, while the numbers of digits of the [[multiplicative inverse|reciprocal]] of 8 and 9 are 2), and the numbers 8 and 9 are in the [[Catalan's conjecture]] (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, the product of 8 and 9 is 60, which is the smallest [[Achilles number]], besides, the concatenation of 8 and 9 is 89, which is the smallest [[Ziesel number]] and the smallest integer such that the factorization of |
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<math>x^n-1</math> |
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over '''Q''' includes coefficients other than |
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<math>\pm 1</math> |
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(i.e. the 89th [[cyclotomic polynomial]], |
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<math>\Phi_{89}</math> |
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, is the first with coefficients other than |
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<math>\pm 1</math> |
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), besides, the squares of 8 and 9 are the only two 2-digit [[automorphic number]]s, besides, 8 and 9 are the only two natural numbers ''n'' such that [[centered polygonal number|centered ''n''-gonal numbers]] (the ''k''th centered ''n''-gonal number is ''n''×''T''<sub>''k''</sub>+1, where ''T''<sub>''k''</sub> is the ''k''th [[triangular number]]) cannot be primes (8 is because all centered 8-gonal numbers are [[square number]]s (4-gonal numbers), 9 is because all centered 9-gonal numbers are [[triangular number]]s (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, in fact, all [[polygonal number]]s with rank > 2 are not primes, i.e. all primes ''p'' cannot be a polygonal number (except the trivial case, i.e. each ''p'' is the second ''p''-gonal number)), assuming the [[Bunyakovsky conjecture]] is true. (i.e. 8 and 9 are the only two natural number ''n'' such that |
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<math>\frac{n}{2}x^2+\frac{n}{2}x+1</math> |
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is not [[Irreducible polynomial|irreducible]]) (Note that for ''n'' = 10, the centered 10-gonal numbers are exactly the [[star number]]s) |
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If m and n are integers ≥3 such that centered m-gonal numbers are always n-gonal numbers, then (m,n) = (8,4) or (9,3), both of these (m,n) pairs have m+n=10. (note that there are no pairs of integers (m,n) with m≥3, n≥3 such that m-gonal numbers are always centered n-gonal numbers) |
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If k=10, then k * prod{ p | k } = 60, which is the smallest [[Achilles number]]. |
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In base 10 ([[dozenal]], the numeral system of this wiki), by [[Midy's theorem]], if 0<''a''<''p'' and the period of the dozenal representation of ''a''/''p'' is 2''n'', so that |
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:<math>\frac{a}{p}=0.\overline{a_1a_2a_3\dots a_na_{n+1}\dots a_{2n}}</math> |
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then the digits in the second half of the repeating dozenal period are the [[Es complement]] of the corresponding digits in its first half (e.g. 1/5 = 0.<span style="text-decoration: overline;">24 97</span>, 1/7 = 0.<span style="text-decoration: overline;">186 X35</span>, and 24 + 97 = EE, 186 + X35 = EEE), and the Es complement of the digits are: |
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* 0 & E (the zero digit & the largest digit) |
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* 1 & X (the unit digit & the only digit whose reciprocal is neither terminating nor purely repeating) |
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* 2/3 & 8/9 (the prime factors of 10 & the difference of "the numbers ''n'' ≤ 10 with terminate reciprocal" and "the divisors of 10" (also the only digits whose reciprocal has exactly 2 digits, also the numbers in the [[Catalan's conjecture]], also the only two natural numbers ''n'' such that [[centered polygonal number|centered ''n''-gonal numbers]] (the ''k''th centered ''n''-gonal number is ''n''×''T''<sub>''k''</sub>+1, where ''T''<sub>''k''</sub> is the ''k''th [[triangular number]]) cannot be primes, see above)) |
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* 4/6 & 5/7 (the only composite proper factors of 10 & the only primes < 10−1 not dividing 10) |
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(note that E is the largest number which is not sum of two composites, and E is also the smallest number >3 which is not sum of two primes) (if you regard 1 as prime, then E is the smallest number >1 which is not sum of two primes) |
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(also, note that the only primes < 10−1 not dividing 10 are 5 and 7 (and the value of 5+7 is exactly 10, and (5,7) are twin primes, and all sums of twin primes ≥(5,7) are also divisible by 10, and 10 is the largest number with this property), and 1/5 = 0.<span style="text-decoration: overline;">24 97</span>, 1/7 = 0.<span style="text-decoration: overline;">186 X35</span>, all digits except 0 and E (the zero digit and the largest digit) appear in the periodic part of either 1/5 or 1/7 (but not both) and exactly once in this number (1/5 or 1/7)) |
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Except 0 and 1, 10 is the only natural number whose square is a Fibonacci number. (i.e. 0, 1 and 100 are the only three [[square number|square]] Fibonacci numbers) |
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If (and only if) there are no Wall-Sun-Sun primes, then 10 is the largest n such that the Pisano period of n<sup>2</sup> is the same as the Pisano period of n (both of them are 20). |
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If n is even and n divides F(n) (where F(n) is the nth Fibonacci number), then n is divisible by 10. (the n’s such that n divides F(n) includes infinitely many multiples of 10, all powers of 5 and other numbers (including some factors of Fib(5^k), e.g. 37501)) |
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The [[Frampton-Kephart prime]]s are the primes ''p'' such that ''p''−1 (or |
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<math>\phi(p)</math> |
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, where |
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<math>\phi</math> |
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is [[Euler's totient function]]) divides 10, these primes are exactly "1 + (the even divisors of 10)" plus the only even prime (2), they (and the number 1) are also the ''k'' such that such that ''d''(''k'') < 3 and ''d''(''k'' − 1) = Pi(''k'') (where ''d''(''k'') the [[divisor count function]] and Pi(''k'') the [[prime counting function]]), they (and the number 1) are also the possible solutions of ''x'' to ''U''(''x'') = ±1 for some [[Lucas sequence]], they are also the prime factors of 63X00, which is the largest number k satisfying that “for any positive integers x,y coprime to k, x^x == y (mod k) iff y^y == x (mod k)”, they are also the prime factors of 95900, the number of vertices of the 20-dimensional Leech lattice. (Note that 95900 = |
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<math>\prod_{p \text{ prime },\ p-1|10}(\text{ if } p|10 \text{ then } p^{6-p},\ \text{ otherwise } p)</math> |
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), and the product of these primes is 16E6, which is the denominator of ''B''<sub>10</sub> (where ''B''<sub>''n''</sub> is the ''n''th [[Bernoulli number]]), (note that ''B''<sub>10</sub> is also the first Bernoulli number with even index whose numerator has a prime factor ''p'' ≥ ''n''+3 (''p''=497) (i.e. 10 is the smallest number ''n'' such that there exists prime ''p'' such that (''p'', ''n'') is an [[irregular pair]])) besides, the dozenal representation of 1/''p'' (with ''p'' over these primes), is terminate when ''p'' divides 10 (i.e. ''p'' = 2 or 3), and for the other three values of ''p'', they are 0.<span style="text-decoration: overline;">2497</span>, 0.<span style="text-decoration: overline;">186X35</span>, and 0.<span style="text-decoration: overline;">0E</span>, the period of them uses all the digits 0, 1, 2, ..., E exactly once. |
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The numbers n have this property: “all primes not divide n are congruent to 1 or −1 mod n” are {1, 2, 3, 4, 6}, they are exactly the proper divisors of 10. (and the LCM of them is 10) |
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The set of the proper divisors of 10 is the complete set of ''k'' in ''N'' such that |
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<math>2\cos\frac{2\pi}{k}</math> |
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is in ''Z''. |
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<span class="">10 is the largest natural number n such that the two sets are completely the same: {a | 0<=a<n, either a=1 or a is prime not dividing n} and {a | 0<=a<n, n+a is prime}. (for n=10, the two sets are both {1, 5, 7, E}, both of the two sets are exactly the natural numbers <10 coprime to 10) (the natural numbers n such that the two sets are completely the same are 1, 2, 4, 6, and 10, all of them are divisors of 10)</span> |
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A number k satisfying that “for any positive integers x,y coprime to k, x^x == y (mod k) iff y^y == x (mod k)” if and only if k is divisible by |
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<math>\lambda(k)</math> |
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(where |
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<math>\lambda(n)</math> |
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is the [[Carmichael lambda function]] and k is a divisor of 63X00, note that 63X00 is the largest k such that |
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<math>\lambda(k)=20</math> |
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, and for all these k’s, |
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<math>\lambda(k)</math> |
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also divides 20, besides, the prime factors of these numbers are exactly the [[Frampton-Kephart prime]]s, i.e. the primes ''p'' such that ''p''−1 (or |
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<math>\phi(p)</math> |
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, where |
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<math>\phi</math> |
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is [[Euler's totient function]]) divides 10. (equivalently, ''p''−1 divides 20, since there is no prime ''p'' such that ''p''−1 divides 20 but ''p''−1 does not divide 10 (both 9 and 21 are composite (9=3<sup>2</sup>, 21=5<sup>2</sup>, both of them are squares of primes)), in fact, the primes ''p'' such that ''p''−1 divides 10 (also 20) are exactly "1 + (the even divisors of 10)" plus the only even prime (2). |
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14060 is the smallest number divisible by all natural numbers from 1 to 10, it is also the smallest number cannot be written in [[primorial base]] using only the dozenal digits (i.e. the digits 0, 1, 2, ..., E) (for [[factorial base]], the smallest such number is 1145000000). |
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10 is the largest known even number expressible as the sum of two primes in only one way (5+7). |
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Primes p such that znorder(Mod(m,p)) = (p-1)/n (i.e. k/p has n different cycles (1 ≤ k ≤ p−1) in [[Radix|base]] m) exist for all (positive or negative) integers m not equal to 0 or ±1 if and only if n is divisible by 10. (if we require that m is positive (and >1), then such primes p exist for all positive integers m>1 if and only if n ends with 0, 2, 6, or X) |
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10 is the smallest (and the only known) ''n''>2 such that there exist ''n'' consecutive integers with ''n'' divisors (at least start with 507711EE945E5199770213X692X2X3, there are 11 consecutive integers with 10 divisors). (The only other two possible such ''n'' are 20 and X0) |
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All numbers ≤10 are panconsummate numbers, but the next number (11) is not, besides, the square of 10 (100) is conjectured to be the largest panconsummate number divisible by either 4 or 6 (i.e. not only for divisible by 10). |
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10 is the smallest ''n''>1 such that the ''n''-th [[cyclotomic polynomial]] has no [[root]] [[mod]] ''p'' for all primes ''p''≤''n''. |
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Consider the prime race mod q (where q≥2) between qn+1 and qn-1. 10 is conjectured to be the n such that the q where qn+1 first takes lead over qn-1 is largest. (such q is known for all n<=1000 except n=10 and n=20) |
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10 is the smallest even number ''n'' such that there exists a prime ''p'' ≥ ''n''+3 (''p'' = 497) such that ''p'' divides the numerator of ''B''<sub>''n''</sub> (where ''B''<sub>''n''</sub> is the ''n''th [[Bernoulli number]]). (i.e. 10 is the smallest number ''n'' such that there exists prime ''p'' such that (''p'', ''n'') is an [[irregular pair]]) (note that the 10th prime (31) is also the smallest irregular prime) |
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10^2−1 (=EE) is the only number which is product of twin primes but not brilliant number, it is also the smallest composite n coprime to 10 (i.e. not divisible by 2 or 3) such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n. |
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10 is the smallest even number n such that “n^k−1 and n^k+1 are always both prime or both composite for every integer k≥1” is conjectured to be true. (for odd number n, n^k−1 and n^k+1 are both even, and hence both composite (if n^k>3)) |
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The ratio of the first two weird numbers (598/5X = E.E39310...) is very close to 10. |
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The only two known base 10 Wieferich primes are 1685 and 5E685, both of them end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. |
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The sum of this 4 [[fifth power]]s is 10<sup>X</sup>, the Xth [[power of 10]] (i.e. 10,000,000,000): 23<sup>5</sup> + 70<sup>5</sup> + 92<sup>5</sup> + E1<sup>5</sup> = 10<sup>X</sup>, this is a counterexample of [[Euler's sum of powers conjecture]], since 10<sup>X</sup> equals 100<sup>5</sup>, it is also a fifth power, but it only require 4 fifth powers (less than the conjectured 5) to add. (Note that 100 is the smallest number whose fifth power is the sum of 4 or less fifth powers) |
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10 is the smallest weight for which a [[cusp form]] exists. This cusp form is the discriminant Δ(''q'') whose Fourier coefficients are given by the [[Ramanujan]] τ-function and which is (up to a constant multiplier) the 20th power of the [[Dedekind eta function]]. This fact is related to a constellation of interesting appearances of the number 10 in mathematics ranging from the value of the [[Riemann zeta function]] at −1 i.e. ζ(−1) = −1/10, the fact that the abelianization of [[SL(2,Z)]] has 10 elements, and even the properties of lattice polygons. |
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10 is the smallest admirable number. |
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10 is a Pell number, a pronic number, and a pentagonal number (5-gonal number). |
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10 is the number of pentominoes (5-ominoes). |
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1/10 = 10% is the property such that a Rubik's Cube is solvable. |
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10 is the [[least common multiple]] of 3 and 4, the number of sides of the first two [[regular polygon]]s (equilateral triangle (3-gon) and square (4-gon)), 10 is also the least common multiple of 4 and 6, the number of faces of the first two [[regular polyhedron]]s (tetrahedron (4-hedron) and cube (6-hedron)). |
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The [[centered polygonal numbers|centered 10-gonal numbers]] (centered dozagonal numbers) are exactly the [[star number]]s. (thus, the star numbers are exactly the numbers obtained as the concatenation of a [[triangular number]] followed by a 1) |
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The known generalized Fermat primes (primes of the form ''b''<sup>2<sup>''n''</sup></sup>+1) with bases ''b''≤10 are: (this number can be prime only if ''b'' is even (since if it is odd then ''b''<sup>2<sup>''n''</sup></sup>+1 is always even, and thus can’t be prime), thus we only consider even bases) |
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{|class="wikitable" |
{|class="wikitable" |
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|+ final two digits of ''n''<sup>''k'' |
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|base (''b'') |
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!{{diagonal split header|''n''|''k''}}!!0!!1!!2!!3!!4!!5!!6!!7!!8!!9!!X!!E!!10!!11!!12!!13!!14!!15!!16!!17!!18!!19!!1X!!1E!!20!!Period |
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|''n'' |
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|- |
|- |
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!0 |
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|2 |
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|01||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00 |
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|0, 1, 2, 3, 4 |
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!1 |
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|- |
|- |
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!1 |
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|4 |
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|01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01||01 |
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|0, 1, 2, 3 |
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!1 |
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|- |
|- |
||
!2 |
|||
|6 |
|||
|01||02||04||08||14||28||54||X8||94||68||14||28||54||X8||94||68||14||28||54||X8||94||68||14||28||54 |
|||
|0, 1, 2 |
|||
!6 |
|||
|- |
|- |
||
!3 |
|||
|8 |
|||
|01||03||09||23||69||83||09||23||69||83||09||23||69||83||09||23||69||83||09||23||69||83||09||23||69 |
|||
|(impossible since 8 is perfect odd power) |
|||
!4 |
|||
|- |
|- |
||
!4 |
|||
|X |
|||
|01||04||14||54||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54 |
|||
|0, 1 |
|||
!3 |
|||
|- |
|||
!5 |
|||
|01||05||21||X5||41||85||61||65||81||45||X1||25||01||05||21||X5||41||85||61||65||81||45||X1||25||01 |
|||
!10 |
|||
|- |
|||
!6 |
|||
|01||06||30||60||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00 |
|||
!1 |
|||
|- |
|||
!7 |
|||
|01||07||41||47||81||87||01||07||41||47||81||87||01||07||41||47||81||87||01||07||41||47||81||87||01 |
|||
!6 |
|||
|- |
|||
!8 |
|||
|01||08||54||68||54||68||54||68||54||68||54||68||54||68||54||68||54||68||54||68||54||68||54||68||54 |
|||
!2 |
|||
|- |
|- |
||
!9 |
|||
|10 |
|||
|01||09||69||09||69||09||69||09||69||09||69||09||69||09||69||09||69||09||69||09||69||09||69||09||69 |
|||
|0 |
|||
!2 |
|||
|- |
|||
!X |
|||
|01||0X||84||E4||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54||54 |
|||
!1 |
|||
|- |
|||
!E |
|||
|01||0E||X1||2E||81||4E||61||6E||41||8E||21||XE||01||0E||X1||2E||81||4E||61||6E||41||8E||21||XE||01 |
|||
!10 |
|||
|- |
|||
!10 |
|||
|01||10||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00 |
|||
!1 |
|||
|- |
|||
!11 |
|||
|01||11||21||31||41||51||61||71||81||91||X1||E1||01||11||21||31||41||51||61||71||81||91||X1||E1||01 |
|||
!10 |
|||
|- |
|||
!12 |
|||
|01||12||44||08||94||X8||54||28||14||68||94||X8||54||28||14||68||94||X8||54||28||14||68||94||X8||54 |
|||
!6 |
|||
|- |
|||
!13 |
|||
|01||13||69||53||69||53||69||53||69||53||69||53||69||53||69||53||69||53||69||53||69||53||69||53||69 |
|||
!2 |
|||
|- |
|||
!14 |
|||
|01||14||94||54||14||94||54||14||94||54||14||94||54||14||94||54||14||94||54||14||94||54||14||94||54 |
|||
!3 |
|||
|- |
|||
!15 |
|||
|01||15||01||15||01||15||01||15||01||15||01||15||01||15||01||15||01||15||01||15||01||15||01||15||01 |
|||
!2 |
|||
|- |
|||
!16 |
|||
|01||16||30||60||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00 |
|||
!1 |
|||
|- |
|||
!17 |
|||
|01||17||61||77||01||17||61||77||01||17||61||77||01||17||61||77||01||17||61||77||01||17||61||77||01 |
|||
!4 |
|||
|- |
|||
!18 |
|||
|01||18||94||68||14||28||54||X8||94||68||14||28||54||X8||94||68||14||28||54||X8||94||68||14||28||54 |
|||
!6 |
|||
|- |
|||
!19 |
|||
|01||19||09||39||69||99||09||39||69||99||09||39||69||99||09||39||69||99||09||39||69||99||09||39||69 |
|||
!4 |
|||
|- |
|||
!1X |
|||
|01||1X||44||E4||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54||94||14||54 |
|||
!3 |
|||
|- |
|||
!1E |
|||
|01||1E||81||5E||41||9E||01||1E||81||5E||41||9E||01||1E||81||5E||41||9E||01||1E||81||5E||41||9E||01 |
|||
!6 |
|||
|- |
|||
!20 |
|||
|01||20||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00||00 |
|||
!1 |
|||
|} |
|} |
||
The period of the final two digits of powers of a number must be a divisor of 10 (= λ(100)). |
|||
Except the case ''b''=8 (which has algebraic factors: 8<sup>2<sup>''n''</sup></sup>+1 = (2<sup>2<sup>''n''</sup></sup>+1) × (4<sup>2<sup>''n''</sup></sup>−2<sup>2<sup>''n''</sup></sup>+1)), the ''n'' for these ''b'' are exactly (the nonnegative integers ≤ ''m'' for ''m'' = 4 to 0) in order, besides, "if ''b''<sup>2<sup>''n''</sup></sup>+1 is composite, then ''b''<sup>2<sup>''n''+1</sup></sup>+1 is also composite" is conjectured to be true for all even ''b''≤10 (but not for ''b'' = next even number (12)). |
|||
More generally, for every ''n''≥2, the period of the final ''n'' digits of powers of a number must be a divisor of 10<sup>''n''−1</sup> (= λ(10<sup>''n''</sup>)). |
|||
There is a known generalized Cullen prime for all bases ''b'' ≤ 10 (but not for ''b'' = 11). (no matter whether you require ''n'' ≥ ''b''−1 or not) |
|||
{|class="wikitable" |
|||
There is a known generalized Woodall prime for all bases ''b'' ≤ 100 (but not for ''b'' = 101). (no matter whether you require ''n'' ≥ ''b''−1 or not) |
|||
|+ [[digital root]] of ''n''<sup>''k'' |
|||
!{{diagonal split header|''n''|''k''}}!!0!!1!!2!!3!!4!!5!!6!!7!!8!!9!!X!!E!!10!!11!!12!!13!!14!!15!!16!!17!!18!!19!!1X!!1E!!20!!Period |
|||
A 10-sided polygon is a dozagon. A 10-faced polyhedron is a dozahedron. Regular cubes (hexahedrons, 6-faced) and octahedrons (8-faced) both have 10 edges, while regular octadozahedrons (18-faced) have 10 vertices. |
|||
Regular 10-hedron (dozahedron) is the only regular polyhedron (Platonic solid) with the most number of sides (5) of its (regular polygon) faces (pentagon), it is the delineation of the universe in Plato's dialogue, and conjectured to be the shape of the universe. (it is also the regular polyhedron (Platonic solid) with the most number (18) of vertices) |
|||
Regular 10-hedron (dozahedron) is the only regular polyhedron (Platonic solid) not corresponding any of the four elements (regular 4-hedron (tetrahedron) = fire, regular 6-hedron (hexahedron) = earth, regular 8-hedron (octahedron) = air, regular 18-hedron (octadozahedron) = water). |
|||
The evenish numbers (numbers coprime to 10 and 10's digit is even) are exactly the numbers ''n'' such that |
|||
<math>\left(\frac{n}{20}\right)=1</math> |
|||
, and the oddish numbers (numbers coprime to 10 and 10's digit is odd) are exactly the numbers ''n'' such that |
|||
<math>\left(\frac{n}{20}\right)=-1</math> |
|||
. (thus, there are no oddish numbers that are squares) |
|||
Two dozagons (10-gons) and one triangle can fill a plane vertex, all solutions using at least one dozagon are {3, 10, 10}, {4, 6, 10}, {3, 3, 4, 10}, and {3, 4, 3, 10}, but only the first two solutions can fill the plane. |
|||
(there are 19 solutions for filling a plane vertex, but only E of them can fill the plane, the solutions are: |
|||
{3, 7, 36}, {3, 8, 20}, {3, 9, 16}, {3, X, 13}, '''{3, 10, 10}''', {4, 5, 18}, '''{4, 6, 10}''', '''{4, 8, 8}''', {5, 5, X}, '''{6, 6, 6}''' |
|||
{3, 3, 4, 10}, {3, 4, 3, 10}, {3, 3, 6, 6}, '''{3, 6, 3, 6}''', {3, 4, 4, 6}, '''{3, 4, 6, 4}''', '''{4, 4, 4, 4}''' |
|||
'''{3, 3, 3, 3, 6}''', '''{3, 3, 3, 4, 4}''', '''{3, 3, 4, 3, 4}''' |
|||
'''{3, 3, 3, 3, 3, 3}''' |
|||
'''bold''' for the solutions that can fill the plane, note that dozagon is the highest regular polygon in convex uniform tiling, i.e. 10 is the largest bold number in the list) |
|||
10 and 18,E27,099,E93,490,727,709,9E9,1X4,841,59E,264,E0X,X13,59X,268,567,863,258,910,E74,98X,682,454 (=(2<sup>X6</sup>)(2<sup>51</sup> − 1)(2<sup>27</sup> − 1)(2<sup>17</sup> − 1)(2<sup>7</sup> − 1)(2<sup>5</sup> − 1)(2<sup>3</sup> − 1)) are the only two known sublime numbers. (A sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number, i.e. a positive integer ''n'' is sublime number if and only if both |
|||
<math>\sigma_0(n)</math> |
|||
and |
|||
<math>\sigma_1(n)</math> |
|||
are perfect numbers (where |
|||
<math>\sigma_k(n)</math> |
|||
is the sum of the ''k''th powers of the divisors of ''n'', i.e. |
|||
<math>\sigma_k(n)=\sum_{d\mid n} d^k</math> |
|||
), e.g. for ''n''=10 we have |
|||
<math>\sigma_0(10)=1^0+2^0+3^0+4^0+6^0+10^0=6</math> |
|||
and |
|||
<math>\sigma_1(10)=1^1+2^1+3^1+4^1+6^1+10^1=24</math> |
|||
, and both 6 and 24 are perfect numbers, thus 10 is a sublime number) |
|||
Although 6 is a divisor of 10, there exists a group of order 10 (A<sub>4</sub>) without a subgroup with order 6, it is the smallest such example (i.e. 10 is the smallest number n such that there exists k dividing n and a group of order n such that this group has no subgroup with order k) |
|||
All orders of non-solvable groups (thus all orders of non-cyclic simple groups) are divisible by either 10 or 18, and all orders of non-solvable groups ≤ 14000 are divisible by 10, (the smallest order of non-solvable groups not divisible by 10 is 14X28) the first two orders of non-cyclic simple groups are 50 and 120, and the [[greatest common divisor]] of them is indeed 10. |
|||
All odd perfect numbers (if exist) end with 1, 09, 39, 69, or 99, and if an odd perfect number ends with 1 (i.e. = 1 mod 10), then it has at least 10 distinct prime factors. |
|||
10 is the kissing number in three dimensions (for a long time, people did not know that whether the answer is 11, this was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory). (In two demensions, it is 6 = 10÷2, and in four dimensions, it is 20 = 10×2. The values of the kissing numbers are only known in 1, 2, 3, 4, 8 and 20 dimensions (note that all such numbers of dimensions are divisors of 20), and the kissing numbers in these numbers of dimensions are in order 2, 6, 10, 20, 180 and 95900, and the 8 dimension case and the 20 dimension case are in order the [[E8 lattice|''E''<sub>8</sub> lattice]] and the [[Leech lattice]], however, there are also the upper bounds and the lower bounds for all other number of dimensions ≤ 20, see the list below, the dimensions in which the kissing number is known are listed in boldface) |
|||
{| class="wikitable" style="text-align: center; margin-left: 40pt;" |
|||
!Dimension |
|||
!Lower<br />bound |
|||
!Upper<br />bound |
|||
|- |
|- |
||
!0 |
|||
|'''1''' |
|||
|1||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0||0 |
|||
|colspan=2 | '''2''' |
|||
!1 |
|||
|- |
|- |
||
!1 |
|||
|'''2''' |
|||
|1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1 |
|||
|colspan=2 | '''6''' |
|||
!1 |
|||
|- |
|- |
||
!2 |
|||
|'''3''' |
|||
|1||2||4||8||5||X||9||7||3||6||1||2||4||8||5||X||9||7||3||6||1||2||4||8||5 |
|||
|colspan=2 | '''10''' |
|||
!X |
|||
|- |
|- |
||
!3 |
|||
|'''4''' |
|||
|1||3||9||5||4||1||3||9||5||4||1||3||9||5||4||1||3||9||5||4||1||3||9||5||4 |
|||
|colspan=2 | '''20''' |
|||
!5 |
|||
|- |
|- |
||
!4 |
|||
|5 |
|||
|1||4||5||9||3||1||4||5||9||3||1||4||5||9||3||1||4||5||9||3||1||4||5||9||3 |
|||
|34 |
|||
!5 |
|||
|38 |
|||
|- |
|- |
||
!5 |
|||
|6 |
|||
|1||5||3||4||9||1||5||3||4||9||1||5||3||4||9||1||5||3||4||9||1||5||3||4||9 |
|||
|60 |
|||
!5 |
|||
|66 |
|||
|- |
|- |
||
!6 |
|||
|7 |
|||
|1||6||3||7||9||X||5||8||4||2||1||6||3||7||9||X||5||8||4||2||1||6||3||7||9 |
|||
|X6 |
|||
!X |
|||
|E2 |
|||
|- |
|- |
||
!7 |
|||
|'''8''' |
|||
|1||7||5||2||3||X||4||6||9||8||1||7||5||2||3||X||4||6||9||8||1||7||5||2||3 |
|||
|colspan=2 | '''180''' |
|||
!X |
|||
|- |
|- |
||
!8 |
|||
|9 |
|||
|1||8||9||6||4||X||3||2||5||7||1||8||9||6||4||X||3||2||5||7||1||8||9||6||4 |
|||
|216 |
|||
!X |
|||
|264 |
|||
|- |
|- |
||
!9 |
|||
|X |
|||
|1||9||4||3||5||1||9||4||3||5||1||9||4||3||5||1||9||4||3||5||1||9||4||3||5 |
|||
|358 |
|||
!5 |
|||
|3X2 |
|||
|- |
|- |
||
!X |
|||
|E |
|||
|1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1 |
|||
|406 |
|||
!2 |
|||
|606 |
|||
|- |
|- |
||
!E |
|||
|10 |
|||
|1||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E |
|||
|5X0 |
|||
!1 |
|||
|951 |
|||
|- |
|- |
||
!10 |
|||
|11 |
|||
|1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1 |
|||
|802 |
|||
!1 |
|||
|1,245 |
|||
|- |
|- |
||
!11 |
|||
|12 |
|||
|1||2||4||8||5||X||9||7||3||6||1||2||4||8||5||X||9||7||3||6||1||2||4||8||5 |
|||
|E1X |
|||
!X |
|||
|1,X13 |
|||
|- |
|- |
||
!12 |
|||
|13 |
|||
|1||3||9||5||4||1||3||9||5||4||1||3||9||5||4||1||3||9||5||4||1||3||9||5||4 |
|||
|1,598 |
|||
!5 |
|||
|2,996 |
|||
|- |
|- |
||
!13 |
|||
|14 |
|||
|1||4||5||9||3||1||4||5||9||3||1||4||5||9||3||1||4||5||9||3||1||4||5||9||3 |
|||
|2,600 |
|||
!5 |
|||
|4,30E |
|||
|- |
|- |
||
!14 |
|||
|15 |
|||
|1||5||3||4||9||1||5||3||4||9||1||5||3||4||9||1||5||3||4||9||1||5||3||4||9 |
|||
|3,116 |
|||
!5 |
|||
|6,4X8 |
|||
|- |
|- |
||
!15 |
|||
|16 |
|||
|1||6||3||7||9||X||5||8||4||2||1||6||3||7||9||X||5||8||4||2||1||6||3||7||9 |
|||
|4,346 |
|||
!X |
|||
|9,710 |
|||
|- |
|- |
||
!16 |
|||
|17 |
|||
|1||7||5||2||3||X||4||6||9||8||1||7||5||2||3||X||4||6||9||8||1||7||5||2||3 |
|||
|6,210 |
|||
!X |
|||
|12,438 |
|||
|- |
|- |
||
!17 |
|||
|18 |
|||
|1||8||9||6||4||X||3||2||5||7||1||8||9||6||4||X||3||2||5||7||1||8||9||6||4 |
|||
|X,0X0 |
|||
!X |
|||
|19,338 |
|||
|- |
|- |
||
!18 |
|||
|19 |
|||
|1||9||4||3||5||1||9||4||3||5||1||9||4||3||5||1||9||4||3||5||1||9||4||3||5 |
|||
|14,060 |
|||
!5 |
|||
|27,708 |
|||
|- |
|- |
||
!19 |
|||
|1X |
|||
|1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1||X||1 |
|||
|24,X60 |
|||
!2 |
|||
|3E,798 |
|||
|- |
|- |
||
!1X |
|||
|1E |
|||
|1||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E||E |
|||
|45,XX6 |
|||
!1 |
|||
|60,000 |
|||
|- |
|- |
||
!1E |
|||
|'''20''' |
|||
|1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1||1 |
|||
|colspan=2 | '''95,900''' |
|||
!1 |
|||
|- |
|||
!20 |
|||
|1||2||4||8||5||X||9||7||3||6||1||2||4||8||5||X||9||7||3||6||1||2||4||8||5 |
|||
!X |
|||
|} |
|} |
||
The period of the digital roots of powers of a number must be a divisor of X (= λ(E)). |
|||
(note that k(1)=2 divides k(2)=3, k(2)=3 divides k(3)=10, k(3)=10 divides k(4)=20, and the three quotients are all primes, however, k(4)=20 does not divide k(5), since 34≤k(5)≤38) |
|||
{|class="wikitable" |
|||
10 is the difference of "the largest number n such that x2 + x + n is prime for all 0≤x≤n−2" (35) and "the largest number n such that 2x2 + n is prime for all 0≤x≤n−1" (25). |
|||
!''n'' |
|||
!possible digital root of an ''n''th power |
|||
10 is the difference of "the smallest number n such that (define an: a0 = 1, for k > 0, ak = (1+a02+a12+...+ak−12)/k) an is not integer" (37) and "the smallest number n such that ʃ0∞(cos(x)cos(x/2)cos(x/3)...cos(x/n)) ≠ π/2" (27). |
|||
|- |
|||
|0 |
|||
10 is the difference of the first two numbers not of the form ''p''<sup>''m''</sup>×''q''<sup>''n''</sup> with ''p'', ''q'' primes, ''m''≥0, ''n''≥0 (36−26). |
|||
|1 |
|||
|- |
|||
10 is the number of [[Latin square]]s of order 3. |
|||
|= 1, 3, 7, 9 (mod X) |
|||
|any number |
|||
10 is the smallest number such that it is unknown whether there are ''n''−1 mutually orthogonal Latin squares of order ''n''. (Note that there cannot be ''n'' mutually orthogonal Latin squares of order ''n'', if ''n'' > 1) |
|||
|- |
|||
|= 2, 4, 6, 8 (mod X) |
|||
10 is the smallest possible order of [[magic square]] with 1 and consecutive primes starting with 3. (the smallest possible order of [[magic square]] with consecutive primes starting with 3 is 2E) |
|||
|1, 3, 4, 5, 9, E |
|||
|- |
|||
10^3 − 6 and (10^3)/2 − 6 are both Giuga numbers (note that 6 is half of 10), the only other Giuga number ≤ 10^4 is 26 (which is exactly the smallest squarefree number with ≥3 distinct prime factors, thus the smallest candidate of Giuga number) |
|||
|= 5 (mod X) |
|||
|1, X, E |
|||
10 (in fact, 10<sup>3</sup>, the [[cube]] of 10) appears in this form of these three [[almost integer]]s related to the three largest [[Heegner number]]s (117, 57 and 37): |
|||
|- |
|||
|> 0 and divisible by X |
|||
* <math>e^{\pi\sqrt{37}}=10^3(9^2-1)^3+520-4.74596469\mathcal{E}07\ldots\times 10^{-4}</math> |
|||
|1, E |
|||
|} |
|||
* <math>e^{\pi\sqrt{57}}=10^3(19^2-1)^3+520-3.\mathcal{E}\mathcal{E}15385\mathcal{X}402\ldots\times 10^{-6}</math> |
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* <math>e^{\pi\sqrt{117}}=10^3(173^2-1)^3+520-6.82\mathcal{X}19322127\ldots\times 10^{-10}</math> |
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(the numbers with [[scientific notation]] are rounded to 10 significant figures) |
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All of them contains 10<sup>3</sup>, besides, the three negative exponents of 10 of them are −10, −6 and −4 (the negative exponent of 10 of the largest number (''e''<sup>π√117</sup>) is −10, etc.), the absolute values of them are exactly 10/''n'' for ''n'' = 1, 2 and 3, besides, the numbers of the consecutive E around the dozenal point are also 10, 6 and 4 (exactly 10/''n'' for ''n'' = 1, 2 and 3), since all of the integer part of these three numbers end with 51E (the reason is all of them are a multiple of 10<sup>3</sup> (=1000) plus the number 520 minus a number <1, see the formulas above), which already has one E before the dozenal point. |
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In music, 10 is the number of [[pitch class]]es in an [[octave]] (an octave has 10 [[semitone]]s), not counting the duplicated (octave) pitch. Also, the total number of major keys, (not counting enharmonic equivalents) and the total number of minor keys (also not counting equivalents). This applies only to 10 tone [[equal temperament]] (10-TET), the most common tuning used today in western influenced music. |
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10 is used for timekeeping, e.g. one year has 10 months, one day has 20 hours, one hour has 50 minutes, and one minute has 50 seconds (all of these numbers are multiples of 10). Also, there are 10 signs of the zodiac ([[astrological sign]]s), and the Chinese use a 10-year cycle for time-reckoning called Earthly Branches ([[Chinese zodiac]]). |
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One foot is 10 inches. |
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10 is the number of numbers on a clock. |
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10 is the number of yellow stars on the Flag of Europe. |
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10 is the largest possible points for rolling two dices. |
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10 appears in the [[natural logarithm]] of [[Levy’s constant]]: |
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<math>\frac{\pi^2}{10\ln 2}</math> |
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. |
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1/10 (the reciprocal of 10) also appears in the [[limit]] of [[Glaisher-Kinkelin constant]]: |
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<math>A=\lim_{n\rightarrow\infty} \frac{K(n+1)}{n^{n^2/2+n/2+1/10} e^{-n^2/4}}</math> |
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. |
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The unit digit of a [[Fibonacci number]] can be any digit except 6 (if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a [[Lucas number]] cannot be 0 or 9 (thus, no Lucas number is divisible by 10), besides, if a Lucas number ends with 2, then it must end with 0002, i.e., this number is congruent to 2 mod 10<sup>4</sup>. |
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The series 1 + 2 + 3 + 4 + ... is divergent, however, we can use the Riemann zeta function to define its sum: −1/10 (= −0.1), since zeta(−1) = −0.1 (thus, 10 + 20 + 30 + 40 + ... (the sum of all positive multiples of 10) = −1). |
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In the following table, ''F''<sub>''n''</sub> is the ''n''-th Fibonacci number, and ''L''<sub>''n''</sub> is the ''n''-th Lucas number. |
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10 is the smallest [[radix|base]] such that the largest known Mersenne prime p such that 2^p−1 is also (Mersenne) prime (X7) does not contain the digit 1 (note that X7 is also the largest n such that there is no k≤n such that n*k+1 and n*k−1 are twin primes, and the smallest k such that n*k+1 and n*k−1 are twin primes for n=X7 is exactly 100, besides, X7 is also the largest known n such that the Mersenne number 2^n−1 and the Wagstaff number (2^n+1)/3 are both primes, The [[New Mersenne Conjecture]] is that “n is of the form 2^k±1 or 4^k±3 (or both)” + “2^n−1 is prime” + “(2^n+1)/3 is prime” is never 2, and X7 is conjectured to be the largest n such that this number is 3, the second-largest such number is 51, and 51 and 51*2 (=X2) are also the second-largest and the third-largest n such that there is no k≤n such that n*k+1 and n*k−1 are twin primes). |
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{|class="wikitable" |
{|class="wikitable" |
||
!''n''||''F''<sub>''n''</sub>||digit root of ''F''<sub>''n''</sub>||''L''<sub>''n''</sub>||digit root of ''L''<sub>''n''</sub>||''n''||''F''<sub>''n''</sub>||digit root of ''F''<sub>''n''</sub>||''L''<sub>''n''</sub>||digit root of ''L''<sub>''n''</sub> |
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!Dozenal digit |
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!0||1||2||3||4||5||6||7||8||9||X||E |
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|- |
|- |
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|1||1||1||1||1||21||37501||5||81101||E |
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!rowspan=4| [[Astrological sign]] ([[Zodiac]]) |
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|Aries||Taurus||Gemini||Cancer||Leo||Virgo||Libra||Scorpio||Sagittarius||Capricorn||Aquarius||Pisces |
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|- |
|- |
||
|2||1||1||3||3||22||5X301||8||111103||7 |
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|style="background: #FF0000"|Fire |
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|style="color: #FFFFFF; background: #A52A2A"|Earth |
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|style="background: #808080"|Air |
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|style="background: #00FFFF"|Water |
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|style="background: #FF0000"|Fire |
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|style="color: #FFFFFF; background: #A52A2A"|Earth |
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|style="background: #808080"|Air |
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|style="background: #00FFFF"|Water |
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|style="background: #FF0000"|Fire |
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|style="color: #FFFFFF; background: #A52A2A"|Earth |
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|style="background: #808080"|Air |
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|style="background: #00FFFF"|Water |
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|- |
|- |
||
|3||2||2||4||4||23||95802||2||192204||7 |
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|Cardinal||Fixed||Mutable||Cardinal||Fixed||Mutable||Cardinal||Fixed||Mutable||Cardinal||Fixed||Mutable |
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|- |
|- |
||
|4||3||3||7||7||24||133E03||X||2X3307||3 |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|Yang |
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|style="color: #FFFFFF;background: #000000"|Yin |
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|- |
|- |
||
|5||5||5||E||E||25||209705||1||47550E||X |
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!rowspan=3| [[Chinese zodiac]] |
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|Rat||Ox||Tiger||Rabbit||Dragon||Snake||Horse||Goat||Monkey||Rooster||Dog||Pig |
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|- |
|- |
||
| |
|6||8||8||16||7||26||341608||E||758816||2 |
||
|- |
|- |
||
|7||11||2||25||7||27||54E111||1||1012125||1 |
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|Mid-Winter||Late Winter||Early Spring||Mid-Spring||Late Spring||Early Summer||Mid-Summer||Late Summer||Early Autumn||Mid-Autumn||Late Autumn||Early Winter |
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|- |
|- |
||
|8||19||X||3E||3||28||890719||1||176X93E||3 |
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![[Solar term|節氣]] ([[Midpoint|中氣]]) |
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|冬至||大寒||雨水||春分||穀雨||小滿||夏至||大暑||處暑||秋分||霜降||小雪 |
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|- |
|- |
||
|9||2X||1||64||X||29||121E82X||2||2780X64||4 |
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![[RGB color wheel]] |
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|style="background: #FF0000"| Red |
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|style="background: #FF7F00"| Orange |
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|style="background: #FFFF00"| Yellow |
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|style="background: #7FFF00"| Chartreuse green |
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|style="background: #00FF00"| Green |
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|style="background: #00FF7F"| Spring green |
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|style="background: #00FFFF"| Cyan |
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|style="background: #007FFF"| Azure |
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|style="background: #0000FF"| Blue |
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|style="background: #7F00FF"| Violet |
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|style="background: #FF00FF"| Magenta |
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|style="background: #FF007F"| Rose |
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|- |
|- |
||
|X||47||E||X3||2||2X||1XE0347||3||432E7X3||7 |
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!rowspan=3| [[Chromatic scale]] |
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|C||C♯/D♭||D||D♯/E♭||E||F||F♯/G♭||G||G♯/A♭||A||A♯/B♭||B |
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|- |
|- |
||
|E||75||1||147||1||2E||310EE75||5||6XE0647||E |
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|0 sharps/0 flats||7 sharps/5 flats||2 sharps/X flats||9 sharps/3 flats||4 sharps/8 flats||E sharps/1 flat||6 sharps/6 flats||1 sharp/E flats||8 sharps/4 flats||3 sharps/9 flats||X sharps/2 flats||5 sharps/7 flats |
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|- |
|- |
||
|10||100||1||22X||3||30||5000300||8||E22022X||7 |
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|Unison||Minor second||Major second||Minor Third||Major Third||Perfect fourth||Tritone||Perfect fifth||Minor sixth||Major sixth||Minor seventh||Major seventh |
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|- |
|- |
||
|11||175||2||375||4||31||8110275||2||16110875||7 |
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!rowspan=3| [[Lepton]] |
|||
|e<sup>−</sup>||ν<sub>e</sub>||μ<sup>−</sup>||ν<sub>μ</sub>||τ<sup>−</sup>||ν<sub>τ</sub>||e<sup>+</sup>||<span style="text-decoration: overline;">ν</span><sub>e</sub>||μ<sup>+</sup>||<span style="text-decoration: overline;">ν</span><sub>μ</sub>||τ<sup>+</sup>||<span style="text-decoration: overline;">ν</span><sub>τ</sub> |
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|- |
|- |
||
|12||275||3||5X3||7||32||11110575||X||25330XX3||3 |
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|colspan=6| particle |
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|colspan=6| antiparticle |
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|- |
|- |
||
|13||42X||5||958||E||33||1922082X||1||3E441758||X |
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|−1''e''||0''e''||−1''e''||0''e''||−1''e''||0''e''||+1''e''||0''e''||+1''e''||0''e''||+1''e''||0''e'' |
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|- |
|- |
||
|14||6X3||8||133E||7||34||2X3311X3||E||6477263E||2 |
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!rowspan=3|[[Quark]] |
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|u||d||c||s||t||b||<span style="text-decoration: overline;">u</span>||<span style="text-decoration: overline;">d</span>||<span style="text-decoration: overline;">c</span>||<span style="text-decoration: overline;">s</span>||<span style="text-decoration: overline;">t</span>||<span style="text-decoration: overline;">b</span> |
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|- |
|- |
||
|15||E11||2||2097||7||35||47551X11||1||X3EE4197||1 |
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|colspan=6| particle |
|||
|colspan=6| antiparticle |
|||
|- |
|- |
||
|16||15E4||X||3416||3||36||75882EE4||1||148766816||3 |
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|+0.8''e''||−0.4''e''||+0.8''e''||−0.4''e''||+0.8''e''||−0.4''e''||−0.8''e''||+0.4''e''||−0.8''e''||+0.4''e''||−0.8''e''||+0.4''e'' |
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|- |
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|17||2505||1||54E1||X||37||101214X05||2||23075X9E1||4 |
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|- |
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|18||3XE9||E||8907||2||38||176X979E9||3||379305607||7 |
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|- |
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|19||6402||1||121E8||1||39||2780E0802||5||5X9X643E8||E |
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|- |
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|1X||X2EE||1||1XE03||3||3X||432E885EE||8||967169X03||7 |
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|- |
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|1E||14701||2||310EE||4||3E||6XE079201||2||13550121EE||7 |
|||
|- |
|||
|20||22X00||3||50002||7||40||E22045800||X||2100180002||3 |
|||
|} |
|} |
||
(Note that ''F''<sub>2X</sub> begins with ''L''<sub>1X</sub>, and ''F''<sub>2E</sub> begins with ''L''<sub>1E</sub>) |
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There are 10 kinds of [[lepton]]s: (including antiparticle) |
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The period of the digit root of Fibonacci numbers is X. |
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(particle): 0 = e<sup>−</sup>, 1 = ν<sub>e</sub>, 2 = μ<sup>−</sup>, 3 = ν<sub>μ</sub>, 4 = τ<sup>−</sup>, 5 = ν<sub>τ</sub> (charge: 2k have −1*''e'', 2k+1 have 0*''e'') |
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The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final ''n'' digits is 2×10<sup>''n''−1</sup> (''n'' ≥ 2). (see [[Pisano period]]) |
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(antiparticle): 6 = e<sup>+</sup>, 7 = <span style="text-decoration: overline;">ν</span><sub>e</sub>, 8 = μ<sup>+</sup>, 9 = <span style="text-decoration: overline;">ν</span><sub>μ</sub>, X = τ<sup>+</sup>, E = <span style="text-decoration: overline;">ν</span><sub>τ</sub> (charge: 2k have +1*''e'', 2k+1 have 0*''e'') |
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There are only 13 possible values (of the totally 100 values, thus only 13%) of the final two digits of a Fibonacci number (see {{oeis|A066853}}). |
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There are 10 kinds of [[quark]]s: (including antiparticle) |
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Except 0 = ''F''<sub>0</sub> and 1 = ''F''<sub>1</sub> = ''F''<sub>2</sub>, the only square Fibonacci number is 100 = ''F''<sub>10</sub> (100 is exactly the square of 10), thus, 10 is the only base such that 100 is a Fibonacci number (since 100 in a base is just the square of this base, and 0 and 1 cannot be the base of numeral system), and thus we can make the near value of the [[golden ratio]]: ''F''<sub>11</sub>/''F''<sub>10</sub> = 175/100 = 1.75 (since the ratio of two connected Fibonacci numbers is close to the golden ratio, as the numbers get large). Besides, the only cube Fibonacci number is 8 = ''F''<sub>6</sub>. |
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(particle): 0 = u, 1 = d, 2 = c, 3 = s, 4 = t, 5 = b (charge: 2k have +0.8*e, 2k+1 have −0.4*e) |
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The recurring dozenal of 1/F(n) terminates if and only if n is a divisor of 10. (10 is exactly the [[radix|base]] of dozenal) |
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(antiparticle): 6 = <span style="text-decoration: overline;">u</span>, 7 = <span style="text-decoration: overline;">d</span>, 8 = <span style="text-decoration: overline;">c</span>, 9 = <span style="text-decoration: overline;">s</span>, X = <span style="text-decoration: overline;">t</span>, E = <span style="text-decoration: overline;">b</span> (charge: 2k have −0.8*e, 2k+1 have +0.4*e) |
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Factors of the Fibonacci numbers: |
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There are 10 colors in [[RGB color wheel]]: |
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F(10) = 100 = 10<sup>2</sup> |
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0 = Red, 1 = Orange, 2 = Yellow, 3 = Chartreuse green, 4 = Green, 5 = Spring green, 6 = Cyan, 7 = Azure, 8 = Blue, 9 = Violet, X = Magenta, E = Rose |
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(F(10) is exactly the square of 10, and F(10) is also the largest Fibonacci number which is square) |
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There are 10 [[astrological sign]]s ([[zodiac]]): |
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F(10−1) = F(E) = 75 |
|||
0 = Aries, 1 = Taurus, 2 = Gemini, 3 = Cancer, 4 = Leo, 5 = Virgo, 6 = Libra, 7 = Scorpio, 8 = Sagittarius, 9 = Capricorn, X = Aquarius, E = Pisces (4k = fire, 4k+1 = earth, 4k+2 = air, 4k+3 = water; 3k = cardinal, 3k+1 = fixed, 3k+2 = mutable; 2k = yang, 2k+1 = yin) (i and j are suitable (for the value of |i−j|): 2,X --> 0,4,8 --> 5,7 --> 6 --> 1,E --> 3,9) |
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F(10+1) = F(11) = 175 |
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75 × 175 = 10001 = 10<sup>4</sup> + 1 |
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The [[Chinese zodiac]]: |
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(10±1 are both primes, and F(10±1) are also both primes) |
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0 = Rat, 1 = Ox, 2 = Tiger, 3 = Rabbit, 4 = Dragon, 5 = Snake, 6 = Horse, 7 = Goat, 8 = Monkey, 9 = Rooster, X = Dog, E = Pig |
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(10<sup>4</sup> + 1 = 10001 is a semiprime, and 10<sup>4</sup> − 1 is a member of betrothed number pair, together with 5600) |
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F(10−2) = F(X) = 47 = 5 × E |
|||
The 10 midpoints of the 20 [[solar term]]s: |
|||
F(10+2) = F(12) = 275 = 25 × 11 |
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5 × 25 = 101 = 10<sup>2</sup> + 1 |
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0 = Dongzhi (Winter solstice, December solstice), 1 = Dahan, 2 = Yushui, 3 = Chunfen (March equinox), 4 = Guyu, 5 = Xiaoman, 6 = Xiazhi (summer solstice, June solstice), 7 = Dashu, 8 = Chushu, 9 = Qiufen (September equinox), X = Shuangjiang, E = Xiaoxue |
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E × 11 = EE = 10<sup>2</sup> − 1 |
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(10±2 are both semiprimes, and F(10±2) are also both semiprimes) |
|||
There are 10 [[pitch]]es in a [[chromatic scale]] ([[pitch class]]): |
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(10<sup>2</sup> ± 1 are both semiprimes, and 5, E, 25, and 11 are all primes) |
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0 = ''C'', 1 = ''C#''/''Db'', 2 = ''D'', 3 = ''D#''/''Eb'', 4 = ''E'', 5 = ''F'', 6 = ''F#''/''Gb'', 7 = ''G'', 8 = ''G#''/''Ab'', 9 = ''A'', X = ''A#''/''Bb'', E = ''B'' ({''k'', ''k''+4, ''k''+7} = ''k'' major chord, {''k'', ''k''+3, ''k''+7} = ''k'' minor chord, {''k'', ''k''+4, ''k''+8} = ''k'' augmented triad, {''k'', ''k''+3, ''k''+6} = ''k'' diminished triad, {''k'', ''k''+4, ''k''+7, ''k''+E} = ''k'' major seventh chord, {''k'', ''k''+3, ''k''+7, ''k''+X} = ''k'' minor seventh chord, {''k'', ''k''+3, ''k''+7, ''k''+E} = ''k'' minor major seventh chord, {''k'', ''k''+4, ''k''+7, ''k''+X} = ''k'' dominant seventh chord, {''k'', ''k''+3, ''k''+6, ''k''+9} = ''k'' diminished seventh chord, {''k'', ''k''+3, ''k''+6, ''k''+X} = ''k'' half-diminished seventh chord) (4k = C+ (also E+, Ab+), 4k+1 = Db+ (also F+, A+), 4k+2 = D+ (also F#+, Bb+), 4k+3 = Eb+ (also G+, B+); 3k = Cdim7 (also Ebdim7, F#dim7, Adim7), 3k+1 = Dbdim7 (also Edim7, Gdim7, Bbdim7), 3k+2 = Ddim7 (also Fdim7, Abdim7, Bdim7)) |
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In n major (or (n+9) minor), the musical note “m” is replaced by “(m−n) mod 10” (e.g. in D major (or b minor), D is Do (C), E is Re (D), F# is Mi (E), G is Fa (F), A is Sol (G), B is La (A), C# is Si (B)) |
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Number of sharps in n major = 7*n (for n minor, it is 7*(n−9) = 7*(n+3)) |
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Number of flats in n major = 5*n (for n minor, it is 5*(n−9) = 5*(n+3)) |
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(Both can take mod 10, however, we usually choose the result ≤6 (=10/2), if ≥10 then take mod 10, and if >6 then take another formula (sharp or flat), its result will be 10−(the result of the original formula)) |
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The digits in 1/5 = {2,4,7,9}, plus the digit 0 (the smallest digit) is the musical notes in the C-major pentatonic scale, plus the digit E (the largest digit) is the musical notes in the G-major pentatonic scale. |
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The digits in 1/7 = {1,3,5,6,8,X}, plus the digit 0 (the smallest digit) is the musical notes in the C#/Db-major diatonic scale, plus the digit E (the largest digit) is the musical notes in the F#/Gb-major diatonic scale. |
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(note that (C,F#/Gb) and (G,C#/Db) are [[tritone]]s, and tritone is equal to 6 semitones, and 6 is equal to half of 10) |
|||
Value of |i−j|: 0 = unison, 1 = minor second, 2 = major second, 3 = minor third, 4 = major third, 5 = perfect fourth, 6 = tritone, 7 = perfect fifth, 8 = minor sixth, 9 = major sixth, X = minor seventh, E = major seventh, 10 = octave |
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(they are the elements in the cyclic group ''Z''<sub>10</sub>) |
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<div style="overflow:auto"> |
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{|class="wikitable" |
{|class="wikitable" |
||
!''n''||2<sup>''n''</sup>||''n''||2<sup>''n''</sup>||''n''||2<sup>''n''</sup>||''n''||2<sup>''n''</sup>||''n''||2<sup>''n''</sup>||''n''||2<sup>''n''</sup> |
|||
|Number of semitones||Name||Exact value in 10-TET||Dozenal value in 10-TET||Just intonation interval||Difference |
|||
|- |
|||
|1||2||21||E2X20X8||41||5317E5804588X8||61||256906X1X93096E8934X8||81||11X12X743504482569888538X0X8||X1||65933E8691303X448E712227X7E11448X8 |
|||
|- |
|||
|2||4||22||1X584194||42||X633XE408E5594||62||4E161183966171E566994||82||238259286X08944E17554X758194||X2||10E667E51626078895E22445393X2289594 |
|||
|- |
|||
|3||8||23||38E48368||43||190679X815XXE68||63||9X302347710323XE11768||83||4744E6551815689X32XX992E4368||X3||21E113XX305013556EX4488X76784556E68 |
|||
|- |
|||
|4||14||24||75X94714||44||361137942E99E14||64||1786046932206479X23314||84||9289E0XX342E15786599765X8714||X4||43X2279860X026XE1E88955931348XE1E14 |
|||
|- |
|||
|5||28||25||12E969228||45||702273685E77X28||65||3350091664410937846628||85||16557X198685X2E350E7730E95228||X5||878453750180519X3E556XE6626959X3X28 |
|||
|- |
|||
|6||54||26||25E716454||46||120452714EE33854||66||66X0163108821673491054||86||30XE3837514E85X6X1E3261E6X454||X6||15348X72X0340X3787XXE19E10516E787854 |
|||
|- |
|||
|7||X8||27||4EE2308X8||47||2408X5229EX674X8||67||111803062154431269620X8||87||619X7472X29E4E9183X6503E188X8||X7||2X695925806818735399X37X20X31E3534X8 |
|||
|- |
|||
|8||194||28||9EX461594||48||48158X457E912994||68||2234061042X886251704194||88||103792925857X9E634790X07X35594||X8||5916E64E41143526X777873841863X6X6994 |
|||
|- |
|||
|9||368||29||17E8902E68||49||942E588E3E625768||69||446810208595504X3208368||89||20736564E4E397E06936181386XE68||X9||E631E09X82286X5193335274835079191768 |
|||
|- |
|||
|X||714||2X||33E5605E14||4X||1685XE55X7E04E314||6X||891420414E6XX0986414714||8X||41270E09X9X773X116703427519E14||XX||1E063X179445518X36666X52946X136363314 |
|||
|- |
|||
|E||1228||2E||67XE00EX28||4E||314E9XXE93X09X628||6E||1562840829E1981750829228||8E||82521X179793278231206852X37X28||XE||3X107833688XX358711118X56918270706628 |
|||
|- |
|- |
||
|10||2454||30||1139X01E854||50||629E799E678179054||70||2E05481457X37432X1456454||90||144X4383373665344624114X5873854||E0||78213467155986E52222358E1634521211054 |
|||
|0||Unison (''C'')||<math>2^{0/10}=2^0=1</math>||1.000000000000||<math>\frac{1}{1}=1</math>||0.0000% |
|||
|- |
|- |
||
|11||48X8||31||2277803E4X8||51||1057E377E1343360X8||71||5X0X9428E3872865828E08X8||91||2898874672710X6890482298E5274X8||E1||1344269122XE751XX44446E5X3068X424220X8 |
|||
|1||Minor second (''C♯''/''D♭'')||<math>2^{1/10}=2^{0.1}=\sqrt[10]{2}</math>||1.086903X21E40||<math>\frac{14}{13}=1.097249724972...</math>|| +0.E8EE% |
|||
|- |
|- |
||
|12||9594||32||4533407X994||52||20E3X733X268670194||72||E8196855X752550E455X1594||92||557552912522191560944575XX52994||E2||26885162459E2X39888891XE86115884844194 |
|||
|2||Major second (''D'')||<math>2^{2/10}=2^{0.2}=\sqrt[6]{2}</math>||1.157745872896||<math>\frac{9}{8}=1.16</math>|| +0.3XX7% |
|||
|- |
|- |
||
|13||16E68||33||8X668139768||53||41X792678515120368||73||1E43714XE92X4XX1X8XE82E68||93||XE2XX5624X44362E01688E2E98X5768||E3||5154X3048E7X58775555639E5022E549488368 |
|||
|3||Minor third (''D♯''/''E♭'')||<math>2^{3/10}=2^{0.3}=\sqrt[4]{2}</math>||1.232E49502549||<math>\frac{6}{5}=1.249724972497...</math>|| +1.3824% |
|||
|- |
|- |
||
|14||31E14||34||159114277314||54||839365134X2X240714||74||3X872299E6589983959E45E14||94||19X598E049888705X03155X5E758E314||E4||X2X986095E38E532XXXE077XX045XX96954714 |
|||
|4||Major third (''E'')||<math>2^{4/10}=2^{0.4}=\sqrt[3]{2}</math>||1.31518811X39E||<math>\frac{5}{4}=1.3</math>||−1.1735% |
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|- |
|- |
||
|15||63X28||35||2E6228532628||55||147670X269858481228||75||79524577E0E577476E7X8EX28||95||378E75X09755520E8062XE8EE2E5X628||E5||185975016EX75XX65999X1339808E99716X9228 |
|||
|5||Perfect fourth (''F'')||<math>2^{5/10}=2^{0.5}=\sqrt[10]{28}</math>||1.40272X544743||<math>\frac{4}{3}=1.4</math>||−0.1E4X% |
|||
|- |
|- |
||
|16||107854||36||5E0454X65054||56||29312185174E4942454||76||136X48E33X1XE32931E395E854||96||735E2E8172XXX41E41059E5EX5XE9054||E6||34E72X031E92E990E77782677415E7723196454 |
|||
|6||Tritone (''F♯''/''G♭'')||<math>2^{6/10}=2^{0.6}=\sqrt{2}</math>||1.4E79170X07E8||<math>\frac{7}{5}=1.497249724972...</math> |
|||
<math>\frac{\mathcal{X}}{7}=1.5186\mathcal{X}35186\mathcal{X}3...</math>||−1.544E%<br />+1.544E% |
|||
|- |
|- |
||
|17||2134X8||37||EX08X990X0X8||57||5662434X329X96848X8||77||271895X67839X65663X76EE4X8||97||126EX5E432599883X820E7XEE8E9E60X8||E7||69E258063E65E761E3334513282EE32463708X8 |
|||
|7||Perfect fifth (''G'')||<math>2^{7/10}=2^{0.7}=\sqrt[10]{\mathcal{X}8}</math>||1.5E90X8E60645||<math>\frac{3}{2}=1.6</math>|| +0.1E4X% |
|||
|- |
|- |
||
|18||426994||38||1E81597618194||58||E104869865797149594||78||52356E91347790E107931EX994||98||251E8EX864E775479441E39EE5E7E0194||E8||117X4E4107E0EE303X6668X26545EX6490721594 |
|||
|8||Minor sixth (''G♯''/''A♭'')||<math>2^{8/10}=2^{0.8}=\sqrt[3]{4}</math>||1.707042186834||<math>\frac{8}{5}=1.724972497249...</math>|| +1.1735% |
|||
|- |
|- |
||
|19||851768||39||3E42E73034368||59||1X20951750E372296E68||79||X46E1E62693361X213663E9768||99||4X3E5E9509E32X936883X77EXEE3X0368||E9||23389X8213X1EX60791115850X8EE90961242E68 |
|||
|9||Major sixth (''A'')||<math>2^{9/10}=2^{0.9}=\sqrt[4]{8}</math>||1.82217X596EX1||<math>\frac{5}{3}=1.8</math>||−1.3824% |
|||
|- |
|- |
||
|1X||14X3314||3X||7X85E26068714||5X||38416X32X1X724571E14||7X||1891X3E051667038427107E7314||9X||987XEE6X17X659671547933E9EX780714||EX||467579442783E90136222E4X195EE61702485E14 |
|||
|X||Minor seventh (''A♯''/''B♭'')||<math>2^{\mathcal{X}/10}=2^{0.\mathcal{X}}=\sqrt[6]{28}</math>||1.946E42776E70||<math>\frac{14}{9}=1.94</math>||−0.3XX7% |
|||
|- |
|- |
||
|1E||2986628||3E||1394EX50115228||5E||74831865839248E23X28||7E||356387X0X3112074852213E2628||9E||17539EE183390E7122X93667E7E9341228||EE||912E36885347E60270445X9836EEE0320494EX28 |
|||
|E||Major seventh (''B'')||<math>2^{\mathcal{E}/10}=2^{0.\mathcal{E}}=\sqrt[10]{1228}</math>||1.X7X0432367E9||<math>\frac{13}{8}=1.\mathcal{X}6</math>||−0.E8EE% |
|||
|- |
|- |
||
|20||5751054||40||2769E8X022X454||60||12946350E476495X47854||80||6E075381862241294X4427X5054||X0||32X77EX346761E2245967113E3E6682454||100||1625X7154X693E0052088E97471EEX0640969E854 |
|||
|10||Octave (''C'')||<math>2^{10/10}=2^1=2</math>||2.000000000000||<math>\frac{2}{1}=2</math>||0.0000% |
|||
|} |
|} |
||
</div> |
|||
For all digits 1 ≤ ''d'' ≤ X (i.e. all digits other than the largest digit (E)), there exists 0 ≤ ''n'' ≤ 20 such that 2<sup>''n''</sup> starts with the digit ''d''. (This is not true for the digit E, the smallest [[power of 2]] starts with the digit E is indeed 2<sup>21</sup> = E2X20X8) |
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even digits: 0, 2, 4, 6, 8, X |
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2<sup>1XE</sup> = 59E18922E81631X39875663E89X853X91E595336X6114815X5X6929933X288E774E479575X628 may be the largest power of 2 not contain the digit 0, it has 65 digits. |
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odd digits: 1, 3, 5, 7, 9, E |
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The number 2<sup>2<sup>9</sup></sup> = 2<sup>368</sup> (see [[power of 2#Powers of two whose exponents are powers of two]]) is very close to [[googol]] (10<sup>100</sup>), since it has EE digits. (thus, the [[Fermat number]] ''F''<sub>9</sub> (=2<sup>2<sup>9</sup></sup>+1) is very close to googol) (it is a [[mathematical coincidence]]: 2<sup>2<sup>9</sup></sup> ≈ 10<sup>10<sup>2</sup></sup>) |
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cardinal digits: 0, 3, 6, 9 |
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[[1729 (number)|1001]] is the first four-digit palindromic number, and it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=1<sup>3</sup> + 10<sup>3</sup>) = 509 + 6E4 (=9<sup>3</sup> + X<sup>3</sup>) (see [[taxicab number]] for other numbers), and it is also the smallest absolute [[Euler pseudoprime]], note that there is no absolute [[Euler-Jacobi pseudoprime]] and no absolute [[strong pseudoprime]]. Since 1001 = 7×11×17, we can use the [[divisibility rule]] of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6''k''+1, 10''k''+1 and 16''k''+1 are all primes, then the product of them must be a [[Carmichael number]] (absolute [[Fermat pseudoprime]]), the smallest case is indeed 1001 (for ''k'' = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3X9). |
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fixed digits: 1, 4, 7, X |
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All values of ''n'' > 45 (45 is the largest ''n'' such that this is not true, note that 45 is also the smallest prime that produces [[prime reciprocal magic square]]) for incrementally largest values of minimal ''x'' > 1 (or minimal ''y'' > 0) satisfying [[Pell's equation]] <math>x^2-ny^2=1</math> end with 1, and the dozens digit of all such values ''n'' > 2X1 are odd. (these values ''n'' are 2, 5, X, 11, 25, 3X, 45, 51, 91, 131, 1E1, 291, 2X1, 2E1, 391, 471, 711, 751, 971, X91, E31, ...) |
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mutable digits: 2, 5, 8, E |
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The [[denominator]] of every nonzero [[Bernoulli number]] (except <math>B_0=1</math> and <math>B_1=-\frac{1}{2}</math>) ends with 6. |
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fire digits: 0, 4, 8 |
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If ''n'' ends with 2 and ''n''/2 is prime (or 1), then the [[denominator]] of the [[Bernoulli number]] <math>B_n</math> is 6 (this is also true for some (but not all) ''n'' ends with X and ''n''/2 is prime). (if the denominator of the Bernoulli number <math>B_n</math> is 6, then ''n'' ends with 2 or X, but ''n''/2 needs not to be prime or 1, the first counterexample is ''n'' = 82, the denominator of the Bernoulli number <math>B_{82}</math> is 6, but 82/2 = 41 = 7<sup>2</sup> is neither prime nor 1) |
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earth digits: 1, 5, 9 |
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[[square root of 2|<math>\sqrt{2}</math>]] is very close to 1.5, since a near-value for <math>\sqrt{2}</math> is 15/10 (=''N''<sub>4</sub>/''P''<sub>4</sub>, where ''N''<sub>''n''</sub> is ''n''th [[NSW number]], and ''P''<sub>''n''</sub> is ''n''th [[Pell number]], ''N''<sub>''n''</sub>/''P''<sub>''n''</sub> is very close to <math>\sqrt{2}</math> when ''n'' is large). Besides, [[square root of 5|<math>\sqrt{5}</math>]] is very close to 2.2X, since a near-value for <math>\sqrt{5}</math> is 22X/100 (= ''L''<sub>10</sub>/''F''<sub>10</sub>, where ''L''<sub>''n''</sub> is ''n''th [[Lucas number]], and ''F''<sub>''n''</sub> is ''n''th [[Fibonacci number]], ''L''<sub>''n''</sub>/''F''<sub>''n''</sub> is very close to <math>\sqrt{5}</math> when ''n'' is large). |
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air digits: 2, 6, X |
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The [[recurring dozenal]] of the [[multiplicative inverse|reciprocal]] of ''n'' terminates if and only if ''n'' is 3-[[smooth number|smooth]] number (or [[harmonic number]]<ref>[https://oeis.org/A003586/a003586.jpg]</ref>) (i.e. ''n'' is regular to 10 if and only if ''n'' is 3-smooth number (or harmonic number)), since the 3-smooth numbers (or the harmonic numbers) are the numbers that evenly divide [[powers of 10]]. The 3-smooth numbers up to 1000 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, 23, 28, 30, 40, 46, 54, 60, 69, 80, 90, X8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, X16, X80, 1000. They are exactly the numbers ''k'' such that <math>\phi(6k)=2k</math>, where <math>\phi</math> is the [[Euler's totient function]]. The sum of the reciprocals of the 3-smooth numbers is equal to 3, i.e. 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = 1 + 0.6 + 0.4 + 0.3 + 0.2 + 0.16 + 0.14 + 0.1 + ... = 3. Brief proof: 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = (Sum_{m>=0} 1/(2^m)) * (Sum_{n>=0} 1/(3^n)) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3. |
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water digits: 3, 7, E |
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The 3-[[smooth number]]s (or the numbers ''n'' such that the [[multiplicative inverse|reciprocal]] of ''n'' terminates) ≤ 10 are 1, 2, 3, 4, 6, 8, 9, and 10, all of these numbers except 8 and 9 are divisors of 10 (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the [[multiplicative inverse|reciprocal]] of all these ''n'' except 8 and 9 are all 1, while the numbers of digits of the [[multiplicative inverse|reciprocal]] of 8 and 9 are 2), and the numbers 8 and 9 are in the [[Catalan's conjecture]] (i.e. 8 and 9 are the only case of two consecutive [[perfect power]]s), besides, (8, 9) is the largest pair of [[regular number]]s which differ by 1, besides, the product of 8 and 9 is 60, which is the smallest [[Achilles number]], besides, the concatenation of 8 and 9 is 89, which is the smallest [[Ziesel number]] and the smallest integer such that the factorization of <math>x^n-1</math> over '''Q''' includes coefficients other than <math>\pm 1</math> (i.e. the 89th [[cyclotomic polynomial]], <math>\Phi_{89}</math>, is the first with coefficients other than <math>\pm 1</math>), besides, the [[repunit]] with length ''k'' (''R''<sub>''k''</sub>) (where ''k'' = the concatenation of ''n'' and the unit (1), i.e. ''k'' = 10''n''+1) is prime for both ''n'' = 8 and ''n'' = 9, and not for any other ''n'' ≤ 1000, besides, the squares of 8 and 9 are the only two 2-digit [[automorphic number]]s, besides, 8 and 9 are the only two natural numbers ''n'' such that [[centered polygonal number|centered ''n''-gonal number]]s (the ''k''th centered ''n''-gonal number is ''n''×''T''<sub>''k''</sub>+1, where ''T''<sub>''k''</sub> is the ''k''th [[triangular number]]) cannot be primes (8 is because all centered 8-gonal numbers are [[square number]]s (4-gonal numbers), 9 is because all centered 9-gonal numbers are [[triangular number]]s (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, in fact, all [[polygonal number]]s with rank > 2 are not primes, i.e. all primes ''p'' cannot be a polygonal number (except the trivial case, i.e. each ''p'' is the second ''p''-gonal number)), assuming the [[Bunyakovsky conjecture]] is true. (i.e. 8 and 9 are the only two natural number ''n'' such that <math>\frac{n}{2}x^2+\frac{n}{2}x+1</math> is not [[Irreducible polynomial|irreducible]]) (Note that for ''n'' = 10, the centered 10-gonal numbers are exactly the [[star number]]s) |
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English numerals of powers of 10: |
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The smallest n≥1 such that 4×60<sup>n</sup>−1 is prime (where 4 is the smallest composite number, and 60 is the smallest Achilles number, note that 4×60<sup>n</sup> is 3-[[smooth number|smooth]] for all n≥1, thus 1/(4×60<sup>n</sup>) terminates in dozenal, and note that 4×60<sup>n</sup> is exactly 10000 (10<sup>4</sup>) when n=2) is 460089, which contains the only four composite 3-[[smooth number|smooth]] digits (4, 6, 8, 9) exactly once and from small to large in order (4 —> 6 —> 8 —> 9), and with two 0’s (zero digits) inside the middle, this prime (4×60<sup>460089</sup>−1) is 78E425-digit prime and proves the generalized Riesel problem base 60 (proves that 205 is the smallest generalized Riesel number in base 60, i.e. 205 is the smallest k such that gcd(k−1, 60−1) = 1 and k×60<sup>n</sup>−1 is composite for all n≥1, note that 205 = 4×60+4+1 = 4×(60+1)+1). |
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1 one |
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The 3-[[smooth number]]s (or the numbers ''n'' such that the [[multiplicative inverse|reciprocal]] of ''n'' terminates) ≤ 20 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, and 20, all of these numbers are divisors of 100, note that the next two 3-smooth numbers (23 and 28) are not divisors of 100 (23 is because it has more prime factors 3 than 100, and 28 is because it has more prime factors 2 than 100) (thus, the numbers of digits of the [[multiplicative inverse|reciprocal]] of all these ''n'' are all ≤2, while the numbers of digits of the [[multiplicative inverse|reciprocal]] of 23 and 28 are 3). |
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10 dozen / onety |
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[[regular polygon|Regular ''n''-gon]] is constructible using [[neusis construction|neusis]], or an angle trisector if and only if the reciprocal of <math>\varphi(n)</math> is terminating number (where <math>\varphi</math> is [[Euler's totient function]]) (i.e. <math>\varphi(n)</math> is 3-[[smooth number|smooth]], or <math>\varphi(n)</math> is regular to 10), thus the ''n'' ≤ 1000 such that regular ''n''-gon is constructible using neusis, or an angle trisector are 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 2X, 2E, 30, 31, 32, 33, 34, 36, 39, 40, 43, 44, 46, 48, 49, 50, 53, 54, 55, 58, 5X, 60, 61, 62, 64, 66, 68, 69, 70, 71, 76, 77, 7E, 80, 81, 86, 88, 89, 90, 91, 93, 94, 96, 99, 9E, X0, X6, X8, XX, E1, E3, E4, E8, 100, 102, 104, 108, 109, 110, 114, 116, 117, 120, 122, 123, 130, 132, 135, 139, 13X, 140, 141, 142, 143, 150, 154, 156, 160, 162, 163, 165, 166, 168, 170, 176, 17X, 180, 183, 187, 190, 193, 194, 195, 197, 198, 1X2, 1X6, 1X8, 1X9, 1E4, 1E9, 200, 203, 204, 208, 214, 216, 220, 223, 228, 22E, 230, 232, 233, 239, 240, 244, 246, 253, 259, 260, 264, 265, 26X, 276, 278, 280, 282, 284, 286, 293, 299, 2X0, 2X8, 2E0, 300, 301, 304, 306, 30X, 310, 314, 31E, 320, 323, 330, 338, 340, 341, 345, 346, 347, 349, 352, 360, 366, 367, 368, 369, 36X, 372, 374, 384, 390, 394, 395, 396, 3X3, 3X8, 3E3, 3E6, 400, 401, 403, 406, 408, 409, 414, 417, 428, 430, 440, 445, 446, 454, 45X, 460, 464, 466, 469, 473, 475, 476, 480, 487, 488, 490, 4X6, 4X7, 4E6, 500, 508, 509, 50X, 518, 519, 530, 534, 537, 539, 540, 541, 543, 544, 548, 549, 550, 566, 576, 57E, 580, 583, 594, 5X0, 5E3, 600, 602, 608, 609, 610, 618, 620, 628, 63X, 640, 646, 660, 669, 671, 674, 680, 682, 685, 689, 68X, 690, 692, 696, 699, 6X4, 6E3, 700, 710, 712, 714, 716, 718, 724, 728, 739, 748, 753, 760, 768, 76X, 770, 773, 781, 786, 794, 7X6, 7E0, 7E1, 800, 801, 802, 806, 810, 814, 816, 828, 832, 839, 853, 854, 860, 86E, 875, 880, 88X, 890, 891, 8X8, 8E1, 8E8, 8EE, 900, 901, 903, 908, 910, 916, 926, 92X, 930, 940, 947, 952, 954, 959, 960, 969, 977, 990, 992, 9X1, 9E0, X00, X03, X13, X14, X16, X17, X18, X19, X23, X34, X36, X60, X68, X72, X76, X79, X80, X82, X83, X86, X88, X8E, X94, X96, XX0, E10, E27, E30, E3X, E40, E43, E46, E55, E68, E69, E80, E99, EX6, 1000 |
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100 gross / hundred |
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If and only if ''n'' is a divisor of 20, then ''m''<sup>2</sup> = 1 mod ''n'' for every integer ''m'' coprime to ''n''. |
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1000 grossiad / thousand |
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If and only if ''n'' is a divisor of 20, then the [[Dirichlet character]]s mod ''n'' are all [[real numbers|real]]. |
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Sets of 3 zeros |
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If and only if ''n'' is a divisor of 20, then ''n'' is divisible by all numbers less than or equal to the [[square root]] of ''n''. |
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(n) (n-illion) (number of zeros: 3n+3) (value: 10<sup>3n+3</sup>) |
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If and only if ''n'' is a divisor of 20, then ''k''−1 is prime for all divisors ''k''>2 of ''n''. |
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1 million 6 (1,000,000) |
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If and only if ''n''+1 is a divisor of 20, then <math>\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}</math> is [[squarefree number|squarefree]] for all 0 ≤ ''k'' ≤ ''n'', i.e. all numbers in the ''n''th row of the [[Pascal's triangle]] are squarefree (the topmost row (i.e. the row which contains only one 1) of the Pascal's triangle is the 0th row, not the 1st row). (Note that all such ''n'' are primes or 1 or 0, and 20 is the largest number ''m'' such that if ''n''+1 is a divisor of ''m'', then ''n'' is prime or 1 or 0, besides, if and only if ''m'' is a divisor of 20, then ''m'' satisfies this condition) |
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2 billion 9 (1,000,000,000) |
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If we only have the numbers 1 to [[20]] (including 1 and 20), then only the primes dividing 10 (i.e. primes ≤3) can be squared, since 5^2 = 21 > 20, and for the numbers such that the [[multiplicative inverse|reciprocal]] of ''n'' terminates, they can only have at most 2 digits (which is the case of 8, 9, 14, 16 and 20), since the numbers with terminate reciprocal with >2 digits, they must be divisible by either 2^5 = 28 or 3^3 = 23, but both are >20. (these ([[prime power]]) numbers are >20: (prime > 3)^(>1), (odd prime)^(>2), 2^(>4)) |
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3 trillion 10 (1,000,000,000,000) |
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20 is the largest integer that is divisible by all natural numbers no larger than its square root. |
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4 quadrillion 13 (1,000,000,000,000,000) |
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The exponents on the right hand side of |
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5 quintillion 16 (1,000,000,000,000,000,000) |
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<math>(1-x)(1-x^2)(1-x^3) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{10} - x^{13} + x^{1\mathcal{X}} + x^{22} - \cdots.</math> |
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6 sextillion 19 (1,000,000,000,000,000,000,000) |
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are exactly the numbers ''n'' such that 20''n''+1 is square. (note that 10 is one of such numbers) |
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20 is the Euler characteristic of a K3 surface. |
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7 septillion 20 (1,000,000,000,000,000,000,000,000) |
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20 is the order of the cyclic group equal to the stable 3-stem in [[homotopy groups of spheres]]: {{pi}}<sub>''n''+3</sub>(''S''<sup>''n''</sup>) = '''Z'''/20'''Z''' for all ''n'' ≥ 5. |
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8 octillion 23 (1,000,000,000,000,000,000,000,000,000) |
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20 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 10, 20 — are exactly those numbers ''n'' for which every invertible element of the [[commutative ring]] [[modular arithmetic|'''Z'''/''n'''''Z''']] is a square root of 1. It follows that the multiplicative group of invertible elements ('''Z'''/20'''Z''')<sup>×</sup> = {±1, ±5, ±7, ±E} is [[group isomorphism|isomorphic]] to the additive group ('''Z'''/2'''Z''')<sup>3</sup>. This fact plays a role in [[monstrous moonshine]]. |
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9 nonillion 26 (1,000,000,000,000,000,000,000,000,000,000) |
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The [[kissing number]] is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) except 6 and 10 |
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X dekrillion 29 (1,000,000,000,000,000,000,000,000,000,000,000) |
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The [[sphere packing|densest packing]] is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) but not the composite divisors of 10 (i.e. 4, 6, 10) |
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E elpillion 30 (1,000,000,000,000,000,000,000,000,000,000,000,000) |
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If xy≤20, then at least one of x and y is a divisor of 10 (this is not true for xy=21, 5×5=21, but 5 is not a divisor of 10). |
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10 dozillion 33 (1,000,000,000,000,000,000,000,000,000,000,000,000,000) |
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[[googol]] (mod ''n'') = [[googolplex]] (mod ''n'') for all 1 ≤ ''n'' ≤ 20 (but not for ''n'' = 21). |
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11 undozillion 36 (etc...) |
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For all numbers ''n'' ≤ 100 (but not for ''n'' = 101, and not for ''n'' = smallest prime > 100 (i.e. 105)), there is ''k'' ≤ 6 such that ''nk''−1 or ''nk''+1 (or both) is prime. (note that for ''n'' = 101, ''k'' = 7, 8 and 9 also not satisfy this condition, the smallest ''k'' satisfying this condition for ''n'' = 101 is X) |
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12 duodozillion 39 |
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20 is the GCD of all Fermat-Wilson quotients. (thus, Fermat-Wilson quotients are never primes) |
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13 tredozillion 40 |
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20 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 20 such spheres form the cells of a 20-cell), for 3-dimensional space, the kissing number is 10, the centers of 10 such spheres form the faces of a 10-hedron. |
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14 quattuordozillion 43 |
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20! is very close to [[Avogadro constant]], which is defined by 1/10 (10%) of the mass of one carbon-10 atom. |
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15 quindozillion 46 |
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20-karat gold is pure, 16-karat gold is 16 parts gold, 6 parts another metal (forming an alloy with 90% gold), 12-karat gold is 12 parts gold, X parts another metal (forming an alloy with 70% gold), and so forth. |
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16 sexdozillion 49 |
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100 is the smallest number with an equal number of deficient and abundant divisors. |
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17 septendozillion 50 |
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"100+''n'' is prime if and only if 100−''n'' is prime" is true for all ''n''≤40 end with 1 or E except ''n'' = 1E (which 100+1E = 11E is prime, but 100−1E = 100−20+1 = 10<sup>2</sup>−2×10+1 = (10−1)<sup>2</sup>, which has algebraic factors and cannot be prime). |
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18 octodozillion 53 |
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Both 10 and 100 are equal to the sum of (the largest Fermat prime below it) and (the largest Mersenne prime below it) (10 = 5+7, 100 = 15+X7). |
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19 novemdozillion 56 |
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10 is the smallest [[radix|base]] such that “the largest known Mersenne prime p such that 2^p−1 is also (Mersenne) prime” (X7) does not contain the digit 1 (note that X7 is also the largest n such that there is no k≤n such that n×k±1 are twin primes, and the smallest k such that n×k±1 are twin primes for n=X7 is exactly 100, besides, X7 is also the largest known n such that the Mersenne number 2^n−1 and the Wagstaff number (2^n+1)/3 are both primes, The [[New Mersenne Conjecture]] is that “n is of the form 2^k±1 or 4^k±3 (or both)” + “2^n−1 is prime” + “(2^n+1)/3 is prime” is never 2, and X7 is conjectured to be the largest n such that this number is 3, the second-largest such number is 51, and 51 and 51×2 (=X2) are also the second-largest and the third-largest n such that there is no k≤n such that n×k±1 are twin primes, 51 is also the largest n such that there is no k≤2n (also k≤4n) such that n×k±1 are twin primes). |
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1X dekradozillion 59 |
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100 followed by 100 1’s is prime. |
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1E elpindozillion 60 |
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100 is the smallest number whose ''n''th power can be written as the sum of (≥2 and <''n'') positive ''n''th powers for some ''n'' (''n''=5, the formula is 100^5 = 23^5 + 70^5 + 92^5 + E1^5, only four 5th powers). |
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20 vigintillion 63 |
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100 is the smallest number whose 5th power can be written as the sum of 4 positive 5th powers, besides, the smallest number whose 5th power can be written as the sum of 5 positive 5th powers is 60, which is exactly half of 100, besides, the smallest number whose 5th power can be written as the sum of 6 positive 5th powers is 10, which is exactly square root of 100 |
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21 unvigintillion 66 |
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100 is the maximum number of steps of numbers < 4X7 (the smallest number that reach a number > 5414 (the record which gotten by the start value 23), namely 100E74) in Collatz sequence (for the numbers 461, 466, 467 and 477). |
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22 duovigintillion 69 |
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All composite numbers ≤200 has at least one prime factor ≤10+1 (this is not true for the next number (201)). |
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23 trevigintillion 70 |
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All composite numbers ≤200 has at least one palindromic prime factor (this is not true for the next number (201)). |
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24 quattuorvigintillion 73 |
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All composite numbers ≤200 has at least one repdigit prime factor (this is not true for the next number (201)). |
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25 quinvigintillion 76 |
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200 = (smallest n such that 195×2^n+1 is prime) (1E3) + (smallest n such that 2^n+195 is prime) (9). (195 is the second-largest known Fermat prime, and the only known Fermat prime which is irregular prime, and the only known Fermat prime p such that p+2 is not prime) (note that for the largest known Fermat prime 31E15, smallest n such that 31E15×2^n+1 is prime is 200−1 (=1EE), and 2+31E15 is prime (in fact, for all known Fermat primes p except 195, 2+p is also prime), thus for p=31E15, (smallest n such that 31E15×2^n+1 is prime) (1EE) + (smallest n such that 2^n+31E15 is prime) (1) is also 200) |
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26 sexvigintillion 79 |
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The exponents on the right hand side of <math>(1-x)(1-x^2)(1-x^3) \cdots = 1 - x - x^2 + x^5 + x^7 - x^{10} - x^{13} + x^{1\mathcal{X}} + x^{22} - \cdots.</math> are exactly the numbers n such that kn+1 is square for k=20. (note that 10 is one of such numbers) |
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27 septenvigintillion 80 |
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All negative-Pell solvable numbers (i.e. numbers ''n'' such that ''x''<sup>2</sup>−''ny''<sup>2</sup> = −1 is solvable) end with negative-Pell solvable digits (i.e. end with 1, 2, 5 or X). |
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28 octovigintillion 83 |
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By [[Benford's law]], the [[probability]] for the leading digit ''d'' (''d'' ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}) occurs (for some sequences, e.g. [[powers of 2]] (1, 2, 4, 8, 14, 28, 54, X8, 194, 368, 714, 1228, 2454, ...) and [[Fibonacci sequence]] (0, 1, 1, 2, 3, 5, 8, 11, 19, 2X, 47, 75, 100, ...)) are: |
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29 novemvigintillion 86 |
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{|class="wikitable" |
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2X dekravigintillion 89 |
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!''d'' |
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!probability |
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!''d'' |
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!probability |
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|- |
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|1 |
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|34.2% |
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|7 |
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|7.9% |
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|- |
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|2 |
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|1E.6% |
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|8 |
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|6.X% |
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|- |
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|3 |
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|14.8% |
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|9 |
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|6.1% |
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|- |
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|4 |
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|10.E% |
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|X |
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|5.6% |
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|- |
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|5 |
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|X.7% |
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|E |
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|5.1% |
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|- |
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|6 |
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|8.E% |
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|} |
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(Note: the [[percentage]] in the list are also in dozenal, i.e. 20% means 0.2 or <math>\frac{20}{100}=\frac{1}{6}</math>, 36% means 0.36 or <math>\frac{36}{100}=\frac{7}{20}</math>, 58.7% means 0.587 or <math>\frac{587}{1000}</math>) |
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2E elpinvigintillion 90 |
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[[Star number]]s are exactly the numbers obtained as the concatenation of a [[triangular number]] followed by 1 (the triangular numbers are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, ..., and the star numbers are 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, ...), thus, all star numbers end with 1. (The star numbers are exactly the [[centered polygonal number|centered 10-gonal number]]s) |
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30 trigintillion 93 |
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The [[Hilbert number]]s are the numbers end with 1, 5 or 9. (i.e. = 1 mod 4) |
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31 untrigintillion 96 |
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The [[Lagado number]]s are the numbers end with 1, 4, 7 or X. (i.e. = 1 mod 3) |
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32 duotrigintillion 99 |
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The smallest two 4-digit [[palindromic number]]s (1001 and 1111) are both [[Ziesel number]]s, they are also the smallest two [[palindromic number]]s which cannot be prime when read in any base, and the smallest 4-digit [[palindromic number]] (1001) is exactly the smallest [[absolute Euler pseudoprime]] and the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=1<sup>3</sup> + 10<sup>3</sup>) = 509 + 6E4 (=9<sup>3</sup> + X<sup>3</sup>), besides, 1111 is also the only known n such that both n and n+1 are Quasi-Carmichael numbers (squarefree composites n with the property that for every prime factor p of n, p+b divides n+b positively with b being any integer besides 0), besides, the smallest number needing >2 steps in the "Reverse and Add!" problem is 6E, and the palindrome at which it stops is exactly 1111 (6E + E6 = 165, 165 + 561 = 706, 706 + 607 = 1111), besides, 1111 is the smallest composite repunit number, besides, the square of 1111 is 1234321, another palindrome. |
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33 tretrigintillion X0 |
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The largest 4-digit number (EEEE) is a member of a [[betrothed number]] pair (its betrothed number is 5600 (also a 4-digit number, note that 5600 is E-[[smooth number|smooth]]), and if we calculate EEEE/''[[greatest common divisor|gcd]]''(EEEE, 5600), we get the 4-digit [[repunit]] (1111)). |
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34 quattuortrigintillion X3 |
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“the smallest (Fermat) pseudoprime to both base 2 and base 3 that is not Carmichael number” (1691) and “conjectured largest panconsummate number” (1961) are both strobogrammatic numbers (the same upside down), and a number written backwards is another number. (these two numbers are both composite, however, the strobogrammatic numbers 160091 and 190061 are primes, also a number written backwards is another number, thus they are also emirps) (of course, the Zeisel numbers 1001 and 1111 are also strobogrammatic numbers) |
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35 quintrigintillion X6 |
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All [[prime number]]s end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or E), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or E (1 and all prime digits that do not divide 10), since all prime numbers other than 2 and 3 are coprime to 10. |
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36 sextrigintillion X9 |
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The density of primes end with 1 is relatively low, but the density of primes end with 5, 7 and E are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or E) |
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37 septentrigintillion E0 |
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Except (3, 5), all [[twin prime]]s end with (5, 7) or (E, 1), and the density of these two types of twin primes are nearly equal. |
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38 octotrigintillion E3 |
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The sum of any pair of twin primes (other than (3, 5)) ends with 0. |
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39 novemtrigintillion E6 |
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If ''n'' ≥ 3 and ''n'' is not divisible by E, then there are infinitely many primes with [[digit sum]] ''n''. |
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3X dekratrigintillion E9 |
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All [[palindromic prime]]s except 11 has an odd number of digits, since all even-digit palindromic numbers are divisible by 11. The palindromic primes below 1000 are 2, 3, 5, 7, E, 11, 111, 131, 141, 171, 181, 1E1, 535, 545, 565, 575, 585, 5E5, 727, 737, 747, 767, 797, E1E, E2E, E6E |
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3E elpintrigintillion 100 |
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All [[lucky number]]s end with digit 1, 3, 7 or 9. |
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40 quadragintillion 103 |
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Except for 3, all [[Fermat prime]]s end with 5. (In fact, there are only 5 known Fermat primes (3, 5, 15, 195 and 31E15) and it is conjectured that there are no more Fermat primes, interestingly, all digits of all known Fermat primes are odd) |
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41 unquadragintillion 106 |
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Except for 3, all [[Mersenne prime]]s end with 7. (Besides, all Mersenne primes except 3 and 7 end with one of the only two 2-digit Mersenne primes (27 and X7)) |
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42 duoquadragintillion 109 |
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Except for 2 and 3, all [[Sophie Germain prime]]s end with 5 or E. |
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43 trequadragintillion 110 |
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Except for 5 and 7, all [[safe prime]]s end with E. |
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44 quattuorquadragintillion 113 |
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A prime ''p'' is [[Gaussian prime]] (prime in the [[Ring (mathematics)|ring]] <math>Z[i]</math>, where <math>i=\sqrt{-1}</math>) if and only if ''p'' ends with 7 or E (or ''p''=3). (i.e. ''p'' = 3 mod 4) |
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45 quinquadragintillion 116 |
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A prime ''p'' is [[Eisenstein prime]] (prime in the [[Ring (mathematics)|ring]] <math>Z[\omega]</math>, where <math>\omega=\frac{-1+\sqrt{3}i}{2}</math>) if and only if ''p'' ends with 5 or E (or ''p''=2). (i.e. ''p'' = 2 mod 3) |
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46 sexquadragintillion 119 |
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A prime ''p'' can be written as ''x''<sup>2</sup> + ''y''<sup>2</sup> if and only if ''p'' ends with 1 or 5 (or ''p''=2). (i.e. ''p'' = 1 or 2 mod 4) |
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47 septenquadragintillion 120 |
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A prime ''p'' can be written as ''x''<sup>2</sup> + 3''y''<sup>2</sup> if and only if ''p'' ends with 1 or 7 (or ''p''=3). (i.e. ''p'' = 0 or 1 mod 3) |
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48 octoquadragintillion 123 |
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A number ''n'' can be written as ''x''<sup>2</sup> + ''y''<sup>2</sup> if and only if all prime factors of the [[squarefree part]] of ''n'' end with 1 or 5 (or 2). |
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49 novemquadragintillion 126 |
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A number ''n'' can be written as ''x''<sup>2</sup> + 3''y''<sup>2</sup> if and only if all prime factors of the [[squarefree part]] of ''n'' end with 1 or 7 (or 3). |
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4X dekraquadragintillion 129 |
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All numbers ≤ 20 [[coprime]] to 10 are either primes or 1 (unit). (this is not true for 21, 21 is the smallest composite coprime to 10) |
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4E elpinquadragintillion 130 |
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All [[full reptend prime]]s end with 5 or 7. (in fact, for all primes ''p'' ≥ 5, (''p''-1)/(the period length of 1/''p'') is odd [[if and only if]] ''p'' is end with 5 or 7, since 10 is a [[quadratic residue|quadratic nonresidue]] mod ''p'' (i.e. <math>\left(\frac{10}{p}\right)=-1</math>, where <math>\left(\frac{m}{n}\right)</math> is the [[Legendre symbol]]) if and only if ''p'' is end with 5 or 7, by [[quadratic reciprocity]], and if 10 is a quadratic residue mod a prime, then 10 cannot be a [[Primitive root modulo n|primitive root]] mod this prime) However, the converse is not true, 17 is not a full reptend prime, since the recurring digits of 1/17 is 0.076E45076E45..., which has only period 6. If and only if ''p'' is a full reptend prime, then the recurring digits of 1/''p'' is [[cyclic number]], e.g. the recurring digits of 1/5 is the cyclic number 2497 (the cyclic permutations of the digits are this number multiplied by 1 to 4), and the recurring digits of 1/7 is the cyclic number 186X35 (the cyclic permutations of the digits are this number multiplied by 1 to 6). The full reptend primes below 1000 are 5, 7, 15, 27, 35, 37, 45, 57, 85, 87, 95, X7, E5, E7, 105, 107, 117, 125, 145, 167, 195, 1X5, 1E5, 1E7, 205, 225, 255, 267, 277, 285, 295, 315, 325, 365, 377, 397, 3X5, 3E5, 3E7, 415, 427, 435, 437, 447, 455, 465, 497, 4X5, 517, 527, 535, 545, 557, 565, 575, 585, 5E5, 615, 655, 675, 687, 695, 6X7, 705, 735, 737, 745, 767, 775, 785, 797, 817, 825, 835, 855, 865, 8E5, 8E7, 907, 927, 955, 965, 995, 9X7, 9E5, X07, X17, X35, X37, X45, X77, X87, X95, XE7, E25, E37, E45, E95, E97, EX5, EE5, EE7. (Note that for the primes end with 5 or 7 below 30 (5, 7, 15, 17, 25 and 27, all numbers end with 5 or 7 below 30 are primes), 5, 7, 15 and 27 are full reptend primes, and since 5×25 = 101 = <math>\Phi_4(10)</math>, the period of 25 is 4, which is the same as the period of 5, and we can use the test of the divisiblity of 5 to test that of 25 (form the alternating sum of blocks of two from right to left), and since 7×17 = E1 = <math>\Phi_6(10)</math>, the period of 17 is 6, which is the same as the period of 7, and we can use the test of the divisiblity of 7 to test that of 17 (form the alternating sum of blocks of three from right to left), thus, 17 and 25 are not full reptend primes, and they are the only two non-full reptend primes end with 5 or 7 below 30) |
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50 quinquagintillion 133 |
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By [[Midy theorem]], if ''p'' is a prime with even period length (let its period length be ''n''), then if we let <math>\frac{a}{p}=0.\overline{a_1a_2a_3...a_n}</math>, then ''a''<sub>''i''</sub> + ''a''<sub>''i''+''n''/2</sub> = E for every 1 ≤ ''i'' ≤ ''n''/2. e.g. 1/5 = 0.249724972497..., and 24 + 97 = EE, and 1/7 = 0.186X35186X35..., and 186 + X35 = EEE, all primes (other than 2 and 3) ≤ 37 except E, 1E and 31 have even period length, thus they can use Midy theorem to get an E-repdigit number, the length of this number is the period length of this prime. (see [[#Recurring_digits|below]] for the recurring digits for 1/''n'' for all ''n'' ≤ 30) |
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51 unquinquagintillion 136 |
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The [[unique prime]]s below 10<sup>60</sup> are E, 11, 111, E0E1, EE01, 11111, 24727225, E0E0E0E0E1, E00E00EE0EE1, 100EEEXEXEE000101, 1111111111111111111, EEEE0000EEEE0000EEEE0001, 100EEEXEE0000EEEXEE000101, 10EEEXXXE011110EXXXE00011, EEEEEEEE00000000EEEEEEEE00000001, EEE000000EEE000000EEEEEE000EEEEEE001, and the period length of their reciprocals are 1, 2, 3, X, 10, 5, 18, 1X, 19, 50, 17, 48, 70, 5X, 68, 53. |
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52 duoquinquagintillion 139 |
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If ''p'' is a safe prime other than 5, 7 and E, then the [[repeating decimal|period length]] of 1/''p'' is (''p''-1)/2. (this is not true for all primes ends with E (other than E itself), the first counterexample is ''p'' = 2EE, where the period length of 1/''p'' is only 37) |
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53 trequinquagintillion 140 |
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There is no full reptend prime ends with 1, since 10 is quadratic residue for all primes end with 1. (if so, then this prime ''p'' is a proper prime (i.e. for the reciprocal of such primes (1/''p''), each digit 0, 1, 2, ..., E appears in the repeating sequence the same number of times as does each other digit (namely, (''p''−1)/10 times)), see [[repeating decimal#Fractions with prime denominators]]) (In fact, not only for base 10 such primes do not exist, for all bases = 0 mod 4 (i.e. bases end with digit 0, 4 or 8), such primes do not exist) |
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54 quattuorquinquagintillion 143 |
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5 and 7 are the only two safe primes which are also full reptend primes, since except 5 and 7, all safe primes end with E, and 10 is quadratic residue for all primes end with E. (if so, then this prime ''p'' produces a stream of ''p''−1 [[pseudo-random numbers|pseudo-random digits]], see [[repeating decimal#Fractions with prime denominators]]) (In fact, not only for base 10 there are only finitely many such primes, of course for square bases (bases of the form ''k''<sup>2</sup>) only 2 may be full reptend prime (if the base is odd), and all odd primes are not full reptend primes, but since all safe primes are odd primes, for these bases such primes do not exist, besides, for the bases of the form 3''k''<sup>2</sup>, only 5 and 7 can be such primes, the proof for these bases is completely the same as that for base 10) |
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55 quinquinquagintillion 146 |
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{|class="wikitable" |
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56 sexquinquagintillion 149 |
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!''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p''||''p''||period length of 1/''p'' |
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|- |
|||
|2||0||111||3||267||266||41E||20E||591||3X||767||766||927||926||E1E||56E||1107||1106||12E5||12E4||14E1||14E||16X7||16X6 |
|||
|- |
|||
|3||0||117||116||271||27||421||63||59E||2XE||76E||395||955||954||E21||570||1115||1114||1301||760||14E5||14E4||16E5||188 |
|||
|- |
|||
|5||4||11E||6E||277||276||427||426||5E1||159||771||132||95E||48E||E25||E24||1125||1124||1317||506||14EE||85E||16E7||16E6 |
|||
|- |
|||
|7||6||125||124||27E||13E||431||109||5E5||5E4||775||774||965||964||E2E||575||112E||675||1337||512||150E||865||1705||398 |
|||
|- |
|||
|E||1||12E||75||285||284||435||434||5E7||66||77E||39E||971||172||E31||116||1135||1134||133E||77E||1517||1XX||1711||493 |
|||
|- |
|||
|11||2||131||76||291||83||437||436||5EE||2EE||785||784||987||32X||E37||E36||114E||685||1345||1344||1521||436||1715||1714 |
|||
|- |
|||
|15||14||13E||7E||295||294||447||446||611||163||791||3X6||995||994||E45||E44||1151||115||1351||166||1525||1524||1727||1726 |
|||
|- |
|||
|17||6||141||20||2X1||150||455||454||615||614||797||796||9X7||9X6||E61||16||1165||1164||1365||1364||1547||1E2||1735||1734 |
|||
|- |
|||
|1E||E||145||144||2XE||155||457||15X||617||206||7X1||138||9XE||4E5||E67||3X2||1167||42||1367||1366||1561||89||1745||1744 |
|||
|- |
|||
|25||4||147||56||2E1||26||45E||22E||61E||30E||7EE||3EE||9E1||9E||E6E||595||1185||1184||136E||795||156E||97||1747||1746 |
|||
|- |
|||
|27||26||157||12||2EE||37||465||464||637||212||801||140||9E5||9E4||E71||596||118E||6X5||1377||106||1577||1576||1751||32X |
|||
|- |
|||
|31||9||167||166||301||90||46E||7||63E||31E||80E||405||9EE||4EE||E91||2E3||1197||472||138E||7X5||157E||89E||1755||1754 |
|||
|- |
|||
|35||34||16E||95||307||102||471||13||647||216||817||816||X07||X06||E95||E94||11X1||6E||1391||3E3||1585||1584||1757||1756 |
|||
|- |
|||
|37||36||171||96||30E||165||481||24||655||654||825||824||X0E||505||E97||E96||11X5||11X4||1395||1394||1587||2X||176E||995 |
|||
|- |
|||
|3E||1E||175||8||315||314||485||44||661||176||82E||415||X11||56||EX5||EX4||11X7||11X6||13X1||7E0||1591||2E6||1781||4E0 |
|||
|- |
|||
|45||44||17E||9E||321||170||48E||245||665||138||835||834||X17||X16||EE5||EE4||11XE||6E5||13X7||536||15XE||8E5||1785||1784 |
|||
|- |
|||
|4E||25||181||X0||325||324||497||496||66E||335||841||84||X27||156||EE7||EE6||11E7||11E6||13E1||13E||15EE||8EE||178E||9X5 |
|||
|- |
|||
|51||13||18E||X5||327||10X||4X5||4X4||675||674||851||14X||X35||X34||1005||1004||1201||700||13E5||13E4||1601||160||1797||1796 |
|||
|- |
|||
|57||56||195||194||32E||175||4E1||9X||687||686||855||854||X37||X36||1011||73||120E||705||1405||1404||1615||1614||17X1||9E0 |
|||
|- |
|||
|5E||2E||19E||XE||33E||17E||4EE||25E||68E||345||85E||42E||X3E||51E||1017||1016||1211||706||1407||326||1621||910||17X5||17X4 |
|||
|- |
|||
|61||30||1X5||1X4||347||46||507||182||695||694||865||864||X41||188||1021||610||121E||70E||1425||1424||1625||1624||17EE||9EE |
|||
|- |
|||
|67||22||1X7||46||34E||2E||511||266||69E||34E||867||2X2||X45||X44||1027||1026||1231||123||142E||815||1635||274||1807||682 |
|||
|- |
|||
|6E||35||1E1||E6||357||11X||517||516||6X7||6X6||871||152||X4E||525||1041||620||123E||71E||1431||286||1647||276||1815||1814 |
|||
|- |
|||
|75||8||1E5||1E4||35E||18E||51E||45||6E1||6E||881||440||X5E||52E||1047||1046||1245||1244||1437||1436||1655||1654||181E||X0E |
|||
|- |
|||
|81||14||1E7||1E6||365||364||527||526||701||360||88E||445||X6E||535||104E||625||1255||114||143E||81E||1657||61X||1825||1824 |
|||
|- |
|||
|85||84||205||204||375||34||531||276||705||704||8X5||98||X77||X76||1051||313||1257||49X||1445||1444||165E||92E||1831||509 |
|||
|- |
|||
|87||86||217||86||377||376||535||534||70E||365||8X7||2E6||X87||X86||1061||16||125E||72E||1457||1456||1667||622||183E||X1E |
|||
|- |
|||
|8E||45||21E||10E||391||1X6||541||54||711||71||8XE||455||X91||283||106E||635||1261||730||1461||38||1671||936||184E||X25 |
|||
|- |
|||
|91||46||221||66||397||396||545||544||71E||36E||8E5||8E4||X95||X94||107E||63E||126E||735||1465||1464||1677||20X||1861||269 |
|||
|- |
|||
|95||94||225||224||3X5||3X4||557||556||721||370||8E7||8E6||X9E||54E||1087||1086||127E||73E||1467||562||167E||93E||1865||1864 |
|||
|- |
|||
|X7||X6||237||92||3XE||1E5||565||564||727||24X||901||230||XX7||376||109E||64E||1281||740||1471||419||1681||140||186E||X35 |
|||
|- |
|||
|XE||55||241||120||3E5||3E4||575||574||735||734||905||198||XXE||555||10E1||329||1295||94||1475||1474||1685||1684||1875||1874 |
|||
|- |
|||
|E5||E4||24E||125||3E7||3E6||577||116||737||736||907||906||XE7||XE6||10E7||10E6||1297||1296||147E||83E||168E||945||1877||146 |
|||
|- |
|||
|E7||E6||251||73||401||100||585||584||745||744||90E||465||XEE||55E||10EE||65E||12X1||75||148E||845||1697||1696||189E||X4E |
|||
|- |
|||
|105||104||255||254||40E||205||587||1XX||747||9X||91E||46E||E11||1X2||1101||220||12X5||2E8||1495||1494||169E||94E||18X1||210 |
|||
|- |
|||
|107||106||25E||12E||415||414||58E||2X5||751||1X3||921||236||E15||228||1105||1104||12X7||12X6||149E||84E||16X1||950||18XE||X55 |
|||
|} |
|||
{|class="wikitable" |
|||
57 septenquinquagintillion 150 |
|||
!period length |
|||
!primes |
|||
!period length |
|||
!primes |
|||
|- |
|||
|1 |
|||
|E |
|||
|11 |
|||
|1E0411, 69X3901 |
|||
|- |
|||
|2 |
|||
|11 |
|||
|12 |
|||
|157, 7687 |
|||
|- |
|||
|3 |
|||
|111 |
|||
|13 |
|||
|51, 471, 57E1 |
|||
|- |
|||
|4 |
|||
|5, 25 |
|||
|14 |
|||
|15, 81, 106X95 |
|||
|- |
|||
|5 |
|||
|11111 |
|||
|15 |
|||
|X9X9XE, 126180EE0EE |
|||
|- |
|||
|6 |
|||
|7, 17 |
|||
|16 |
|||
|E61, 1061 |
|||
|- |
|||
|7 |
|||
|46E, 2X3E |
|||
|17 |
|||
|1111111111111111111 |
|||
|- |
|||
|8 |
|||
|75, 175 |
|||
|18 |
|||
|24727225 |
|||
|- |
|||
|9 |
|||
|31, 3X891 |
|||
|19 |
|||
|E00E00EE0EE1 |
|||
|- |
|||
|X |
|||
|E0E1 |
|||
|1X |
|||
|E0E0E0E0E1 |
|||
|- |
|||
|E |
|||
|1E, 754E2E41 |
|||
|1E |
|||
|3E, 78935EX441, 523074X3XXE |
|||
|- |
|||
|10 |
|||
|EE01 |
|||
|20 |
|||
|141, 8E5281 |
|||
|} |
|||
The period level of a prime ''p'' ≥ 5 is (''p''−1)/(period length of 1/''p''), e.g., <math>\frac{1}{17}</math> has period level 3, thus the numbers <math>\frac{a}{17}</math> with integer 1 ≤ ''a'' ≤ 16 from 3 different cycles: 076E45 (for ''a'' = 1, 7, 8, E, 10, 16), 131X8X (for ''a'' = 2, 3, 5, 12, 14, 15) and 263958 (for ''a'' = 4, 6, 9, X, 11, 13). Besides, <math>\frac{1}{15}</math> has period level 1, thus this number is a cyclic number and 15 is a full-reptend prime, and all of the numbers <math>\frac{a}{15}</math> with integer 1 ≤ ''a'' ≤ 14 from the cycle 08579214E36429X7. |
|||
58 octoquinquagintillion 153 |
|||
There are only 9 [[repunit prime]]s below ''R''<sub>1000</sub>: ''R''<sub>2</sub>, ''R''<sub>3</sub>, ''R''<sub>5</sub>, ''R''<sub>17</sub>, ''R''<sub>81</sub>, ''R''<sub>91</sub>, ''R''<sub>225</sub>, ''R''<sub>255</sub> and ''R''<sub>4X5</sub> (''R''<sub>''n''</sub> is the repunit with length ''n''). If ''p'' is a [[Sophie Germain prime]] other than 2, 3 and 5, then ''R''<sub>''p''</sub> is divisible by 2''p''+1, thus ''R''<sub>''p''</sub> is not prime. (The length for the repunit (probable) primes are 2, 3, 5, 17, 81, 91, 225, 255, 4X5, 5777, 879E, 198E1, 23175, 311407, ..., note that 879E is the smallest (and the only known) such number ends with E) |
|||
59 novemquinquagintillion 156 |
|||
A prime ''p'' divides ''R''<sub>''p''</sub>−1 (where ''R''<sub>''n''</sub> is the [[repunit]] with length ''n'') if and only if ''p''<sup>2</sup> (the [[square]] of ''p'') is not [[Brazilian number]] (numbers which are nontrivial [[repdigit]] to some [[Radix|base]], these numbers are also highly related to the generalized [[repunit]] numbers). (E is the only prime ''p'' not dividing ''R''<sub>''p''</sub>−1, E is also the only prime ''p'' such that ''p''<sup>2</sup> is Brazilian number (E<sup>2</sup> = X1 = 11111<sub>3</sub>)) |
|||
5X dekraquinquagintillion 159 |
|||
The [[repunit]] number 111 is exactly the largest known number ''n'' such that (''n''<sup>''n''</sup>+1)/(''n''+1) (the generalized [[repunit]] number with length ''n'' in [[negative base]] −''n'') is (probable) prime. |
|||
5E elpinquinquagintillion 160 |
|||
By [[Fermat's little theorem]], if ''p'' is a prime other than 2, 3 and E, then ''p'' divides the [[repunit]] with length ''p''−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54, the counterexamples up to 1000 are 55, 77, E1, 101, 187, 275, 4X7, 777, 781, E55, they are exactly the [[Fermat pseudoprime]]s for base 10 (composite numbers ''c'' such that 10<sup>''c''-1</sup> = 1 mod ''c'') which are not divisible by E, they are called "deceptive primes", if ''n'' is deceptive prime, then ''R''<sub>''n''</sub> is also deceptive prime, thus there are infinitely may deceptive primes) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's. |
|||
60 sexagintillion 163 |
|||
{|class="wikitable" |
|||
61 unsexagintillion 166 |
|||
|+ smallest multiple of ''n'' uses only 0's and 1's |
|||
!''n''||+1||+2||+3||+4||+5||+6||+7||+8||+9||+X||+E||+10 |
|||
|- |
|||
!0+ |
|||
|1||10||10||10||101||10||1001||100||100||1010||11111111111||10 |
|||
|- |
|||
!10+ |
|||
|11||10010||1010||100||10111||100||1001||1010||10010||111111111110||11101||100 |
|||
|- |
|||
!20+ |
|||
|110111||110||1000||10010||101||1010||101011||1000||111111111110||101110||101101||100 |
|||
|- |
|||
!30+ |
|||
|1001001||10010||110||10100||111001||10010||101001||111111111110||10100||111010||10001111||100 |
|||
|- |
|||
!40+ |
|||
|10111101||1101110||101110||110||1100101||1000||1011101111111||100100||10010||1010||101011||1010 |
|||
|- |
|||
!50+ |
|||
|10010101||1010110||100100||1000||1111||111111111110||1100101||101110||111010||1011010||1100111||100 |
|||
|- |
|||
!60+ |
|||
|10101101||10010010||1101110||10010||1011101111111||110||10101011||10100||10000||1110010||100111001||10010 |
|||
|- |
|||
!70+ |
|||
|1101001||1010010||1010||1111111111100||10001||10100||1001||111010||1010110||100011110||101101||1000 |
|||
|- |
|||
!80+ |
|||
|111011||101111010||1111111111100||1101110||110001||101110||10100111||1100||1011010||11001010||100111||1000 |
|||
|- |
|||
!90+ |
|||
|1010111111||10111011111110||10010010||100100||10101001||10010||110101001||1010||1100||1010110||101100011||10100 |
|||
|- |
|||
!X0+ |
|||
|111111111101||100101010||1110010||1010110||11100001||100100||1100001||10000||1010010||11110||1111011111||111111111110 |
|||
|- |
|||
!E0+ |
|||
|1001||11001010||101000||1011100||101011||111010||11010111||1011010||100011110||11001110||1111111111111111111111||100 |
|||
|} |
|||
{|class="wikitable" |
|||
62 duosexagintillion 169 |
|||
|''n'' |
|||
|1 |
|||
|5 |
|||
|7 |
|||
|E |
|||
|11 |
|||
|15 |
|||
|17 |
|||
|1E |
|||
|21 |
|||
|25 |
|||
|27 |
|||
|2E |
|||
|- |
|||
|smallest ''k'' such that ''k''×''n'' is a repunit |
|||
|1 |
|||
|275 |
|||
|1X537 |
|||
|123456789E |
|||
|1 |
|||
|92X79E43715865 |
|||
|8327 |
|||
|69E63848E |
|||
|634X159788253X72E1 |
|||
|55 |
|||
|509867481E793XX5X1243628E317 |
|||
|45X3976X7E |
|||
|- |
|||
|the length of the repunit ''k''×''n'' |
|||
|1 |
|||
|4 |
|||
|6 |
|||
|E |
|||
|2 |
|||
|14 |
|||
|6 |
|||
|E |
|||
|18 |
|||
|4 |
|||
|26 |
|||
|10 |
|||
|} |
|||
(this ''k'' is usually not prime, in fact, this ''k'' is not prime for all numbers ''n'' < 100 which are coprime to 10 except ''n'' = 55, and for ''n'' < 1000 which is coprime to 10, this ''k'' is prime only for ''n'' = 55, 101, 19E, 275 and 46E, and only 19E and 46E are itself prime, other 3 numbers are 5×11, 5×25 and 11×25, and this ''k'' for these ''n'' are successively 25, 11 and 5, which makes ''k''×''n'' = ''R''<sub>4</sub> = 1111 = 5×11×25, besides, this ''k'' for ''n'' = 46E is 2X3E, which makes ''k''×''n'' = ''R''<sub>7</sub> = 1111111, a repunit [[semiprime]], and this ''k'' for ''n'' = 19E is a X8-digit prime number, with ''k''×''n'' = ''R''<sub>XE</sub>, another repunit semiprime) |
|||
63 tresexagintillion 170 |
|||
For every prime ''p'' except E, the repunit with length ''p'' is congruent to 1 mod ''p''. (The converse is also not true, the counterexamples up to 1000 are 4, 6, 10, 33, 55, 77, E1, 101, 187, 1E0, 275, 444, 4X7, 777, 781, E55, they are called "repunit pseudoprimes" (or weak deceptive primes), all deceptive primes are also repunit pseudoprimes, if ''n'' is repunit pseudoprime, then ''R''<sub>''n''</sub> is also repunit pseudoprime, thus there are infinitely may repunit pseudoprimes. No repunit pseudoprimes are divisible by 8, 9 or E. (in fact, the repunit pseudoprimes are exactly the weak pseudoprimes for base 10 (composite numbers ''c'' such that 10<sup>''c''</sup> = 10 mod ''c'') which are not divisible by E) Besides, the deceptive primes are exactly the repunit pseudoprimes which are coprime to 10) |
|||
64 quattuorsexagintillion 173 |
|||
Smallest multiple of ''n'' with digit sum 2 are: (0 if not exist) |
|||
65 quinsexagintillion 176 |
|||
:2, 2, 20, 20, 101, 20, 1001, 20, 200, 1010, 0, 20, 11, 10010, 1010, 200, 100000001, 200, 1001, 1010, 10010, 0, 0, 20, 10000000001, 110, 2000, 10010, 101, 1010, 1000000000000001, 200, 0, 1000000010, 1000000000001, 200, ..., if and only if ''n'' is divisible by some prime ''p'' with 1/''p'' odd period length, then such number does not exist. |
|||
Smallest multiple of ''n'' with digit sum 3 are: (0 if not exist) |
|||
66 sexsexagintillion 179 |
|||
:3, 12, 3, 30, 21, 30, 12, 120, 30, 210, 0, 30, 0, 12, 210, 300, 201, 30, 10101, 210, 120, 0, 1010001, 120, 21, 0, 300, 120, 0, 210, 1010001, 1200, 0, 2010, 200001, 30, ..., such number does not exist for ''n'' divisible by E, 11 or 25. |
|||
Smallest multiple of ''n'' with digit sum 4 are: (0 if not exist) |
|||
67 septensexagintillion 180 |
|||
:4, 4, 13, 4, 13, 40, 103, 40, 130, 130, 0, 40, 22, 1030, 13, 40, 3001, 130, 2002, 130, 103, 0, 11101, 40, 10012, 22, 1300, 1030, 202, 130, 10003, 400, 0, 30010, 101101, 130, ..., such number is conjectured to exist for all ''n'' not divisible by E (of course, if ''n'' is divisible by E, then such number does not exist). |
|||
Smallest multiple of ''n'' with digit sum ''n'' are: |
|||
68 octosexagintillion 183 |
|||
:1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 1E0, 20E, 22X, 249, 268, 287, 2X6, 45X, 488, 4E6, 1EX, 8E4, 3EX0, 3EE, 23EX, 1899, XX8, 2E79, 4E96, 1EX9, 4XX8, 2EE9, 3XEX, 799X, 5EE90, ..., such number is conjectured to exist for all ''n''. |
|||
45 is the smallest prime that produces [[prime reciprocal magic square]], i.e. write the [[repeating decimal|recurring digits]] of 1/45 (=0.{{overline|2872E3X23205525X784640XX4E9349081989E6696143757E117}}, which has period 44) to 44/45, we get a 44×44 [[prime reciprocal magic square]] (its magic number is 1EX), it is conjectured that there are infinitely many such primes, but 45 is the only such prime below 1000, all such primes are [[full reptend prime]]s, i.e. the reciprocal of them are [[cyclic number]]s, and 10 is a [[Primitive root modulo n|primitive root]] modulo these primes. |
|||
69 novemsexagintillion 186 |
|||
All numbers of the form 34{1} are composite (proof: 34{1<sub>''n''</sub>} = 34×10<sup>''n''</sup>+(10<sup>''n''</sup>−1)/E = (309×10<sup>''n''</sup>−1)/E and it can be factored to ((19×10<sup>''n''/2</sup>−1)/E) × (19×10<sup>''n''/2</sup>+1) for even ''n'' and divisible by 11 for odd ''n''). Besides, 34 was proven to be the smallest ''n'' such that all numbers of the form ''n''{1} are composite. However, the smallest prime of the form 23{1} is 23{1<sub>E78</sub>}, it has E7X digits. The only other two ''n''≤100 such that all numbers of the form ''n''{1} are composite are 89 and 99 (the reason of 89 is the same as 34, and the reason of 99 is 99{1<sub>''n''</sub>} is divisible by 5, 11 or 25). |
|||
6X dekrasexagintillion 189 |
|||
The only known of the form 1{0}1 is 11 (see [[generalized Fermat prime]]), these are the primes obtained as the concatenation of a power of 10 followed by a 1. If ''n'' = 1 mod 11, then all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are divisible by 11 and thus composite. Except 10, the smallest ''n'' not = 1 mod 11 such that all numbers obtained as the concatenation of a power of ''n'' (>1) followed by a 1 are composite was proven by EX, since all numbers obtained as the concatenation of a power of EX (>1) followed by a 1 are divisible by either E or 11 and thus composite. However, the smallest prime obtained as the concatenation of a power of 58 (>1) followed by a 1 is 10×58<sup>2781E5</sup>+1, it has 459655 digits. |
|||
6E elpinsexagintillion 190 |
|||
All numbers of the form 1{5}1 are composite (proof: 1{5<sub>''n''</sub>}1 = (14×10<sup>''n''+1</sup>−41)/E and it can be factored to (4×10<sup>(''n''+1)/2</sup>−7) × ((4×10<sup>(''n''+1)/2</sup>−7)/E) for odd ''n'' and divisible by 11 for even ''n''). |
|||
70 septuagintillion 193 |
|||
The [[emirp]]s below 1000 are 15, 51, 57, 5E, 75, E5, 107, 117, 11E, 12E, 13E, 145, 157, 16E, 17E, 195, 19E, 1X7, 1E5, 507, 51E, 541, 577, 587, 591, 59E, 5E1, 5EE, 701, 705, 711, 751, 76E, 775, 785, 7X1, 7EE, E11, E15, E21, E31, E61, E67, E71, E91, E95, EE5, EE7. |
|||
71 unseptuagintillion 196 |
|||
The non-repdigit [[permutable prime]]s below [[googolplex|10<sup>10<sup>100</sup></sup>]] are 15, 57, 5E, 117, 11E, 5EEE (the smallest representative prime of the permutation set). |
|||
72 duoseptuagintillion 199 |
|||
The non-repdigit [[circular prime]]s below [[googolplex|10<sup>10<sup>100</sup></sup>]] are 15, 57, 5E, 117, 11E, 175, 1E7, 157E, 555E, 115E77 (the smallest representative prime of the cycle). |
|||
73 treseptuagintillion 1X0 |
|||
The first few [[Smarandache prime]]s are the concatenation of the first 5, 15, 4E, 151, ... positive integers. |
|||
74 quattuorseptuagintillion 1X3 |
|||
The only known [[Smarandache–Wellin prime]]s are 2 and 2357E11. |
|||
75 quinseptuagintillion 1X6 |
|||
There are 15 [[minimal prime (recreational mathematics)|minimal prime]]s, and they are 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001. |
|||
76 sexseptuagintillion 1X9 |
|||
The smallest [[weakly prime]] is 6E8XE77. |
|||
77 septenseptuagintillion 1E0 |
|||
There are 824E1 left-[[truncatable prime]]s, the largest of them is 28-digit 471X34X164259EX16E324XE8X32E7817. |
|||
78 octoseptuagintillion 1E3 |
|||
There are 12E right-[[truncatable prime]]s, the largest of them is X-digit 375EE5E515. |
|||
79 novemseptuagintillion 1E6 |
|||
There are 2E primes which are both left-truncatable and right-truncatable. They have been called [[two-sided primes]], and they are 2, 3, 5, 7, E, 25, 27, 35, 37, 3E, 57, 5E, 75, E5, E7, 25E, 315, 357, 35E, 375, 3E5, 3E7, 517, 51E, 575, 5E5, 5E7, 2557, 35E7, 5117, 511E, 51E7, 5E1E, 375E5, 511E7. |
|||
7X dekraseptuagintillion 1E9 |
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The only two [[Wieferich prime#Base-a Wieferich primes|base 10 Wieferich prime]]s up to 10<sup>10</sup> are 1685 and 5E685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. (there is also a note for the only two known base 2 [[Wieferich prime]]s (771 and 2047) minus 1 written in base [[binary number|2]], [[octal|8]] (= 2<sup>3</sup>) and [[hexadecimal|14]] (= 2<sup>4</sup>), 770 = 010001000100<sub>(2)</sub> = 444<sub>(14)</sub> is a [[repdigit]] in base 14, and 2046 = 110110110110<sub>(2)</sub> = 6666<sub>(8)</sub> is also a repdigit in base 8, see [[Wieferich prime#Binary periodicity of p − 1]]) |
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7E elpinseptuagintillion 200 |
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For the numbers between 5X0 (the smallest number divisible by all of the numbers 1 to 8) and 630 (the square of 26 = 5#) and end with 1, 5, 7 or E (the digits coprime to 10), all numbers whose dozen digit is odd are primes, and all numbers whose dozen digit is even are composites. |
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80 octogintillion 203 |
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For all odd [[composite number|composite]]s ''c'' up to 1000, there exists integer ''a'' such that [[greatest common divisor|GCD]](''a'', ''c'') = 1 and ''a''<sup>(''c''−1)/2</sup> is congruent to neither 1 nor −1 mod ''c'' (i.e. ''c'' is not an [[Euler pseudoprime]] base ''a''), however, this is not true for ''c'' = [[1001]], 1001 is Euler pseudoprime to all bases [[coprime]] to itself, i.e. 1001 is an [[absolute Euler pseudoprime]]. (note that there are no “absolute Euler-Jacobi pseudoprime”, every composite number is Euler-Jacobi pseudoprime to at most 30% (i.e. 1/4) bases coprime to it) |
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81 unoctogintillion 206 |
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There are 1, 2, 3, 5 and 6-digit (but not 4-digit) [[narcissistic number]]s, there are totally 73 narcissistic numbers, the first few of which are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, ..., the largest of which is 43-digit 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E. (see {{oeis|A161949}}) |
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82 duooctogintillion 209 |
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The only two [[factorion]]s are 1 and 2. |
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83 treoctogintillion 210 |
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The only seven [[happy number]]s below 1000 are 1, 10, 100, 222, 488, 848 and 884, almost all natural numbers are unhappy. All unhappy numbers get to one of these four cycles: {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {68, 84}, or one of the only two fixed points other than 1: 25 and X5. |
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84 quattuoroctogintillion 213 |
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If we use the sum of the cubes (instead of squares) of the digits, then every natural numbers get to either 1 or the cycle {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}. (for the example of the famous Hardy–Ramanujan number 1001 = 9<sup>3</sup> + X<sup>3</sup>, we know that this sequence with initial term 9X is 9X, 1001, 2, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, ...) |
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85 quinoctogintillion 216 |
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{|class="wikitable" |
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86 sexoctogintillion 219 |
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!''n'' |
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!fixed points and cycled for the sequence for sum of ''n''-th powers of the digits |
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!length of these cycles |
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|- |
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!1 |
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|{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {X}, {E} |
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|1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 |
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|- |
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!2 |
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|{1}, {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {25}, {68, 84}, {X5} |
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|1, 2, X, 3, 1, 2, 1 |
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|- |
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!3 |
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|{1}, {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}, {577}, {668}, {6E5, E74, 100X}, {X83}, {11XX} |
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|1, 12, 1, 1, 3, 1, 1 |
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|- |
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!4 |
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|{1}, {X6X, 103X8, 8256, 35X9, 9EXE, 22643, E69, 1102X, 596X, X842, 8394, 6442, 1080, 2455}, {206X, 6668, 4754}, {3X2E, 12396, 472E, X02X, E700, 9X42, 98X9, 13902} |
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|1, 12, 3, 8 |
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|} |
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The [[harshad number]]s up to 200 are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 1X, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, X0, X1, E0, 100, 10X, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1X0, 1E0, 1EX, 200, although the sequence of [[factorial]]s begins with harshad numbers, not all factorials are harshad numbers, after 7! (=2E00, with digit sum 11 but 11 does not divide 7!), 8X4! is the next that is not (8X4! has digit sum 8275 = E×8E7, thus not divide 8X4!). There are no 21 consecutive integers that are all harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all harshad numbers. |
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87 septenoctogintillion 220 |
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The [[Kaprekar number]]s up to 10000 are 1, E, 56, 66, EE, 444, 778, EEE, 12XX, 1640, 2046, 2929, 3333, 4973, 5E60, 6060, 7249, 8889, 9293, 9E76, X580, X912, EEEE. |
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88 octooctogintillion 223 |
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The [[Kaprekar's routine]] of any four-digit number which is not [[repdigit]] converges to either the cycle {3EE8, 8284, 6376} or the cycle {4198, 8374, 5287, 6196, 7EE4, 7375}, and the Kaprekar map of any three-digit number which is not repdigit converges to the fixed point 5E6, and the Kaprekar map of any two-digit number which is not repdigit converges to the cycle {0E, X1, 83, 47, 29, 65}. |
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89 novemoctogintillion 226 |
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{|class="wikitable" |
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8X dekraoctogintillion 229 |
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!''n'' |
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!Cycles for Kaprekar's routine for ''n''-digit numbers |
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8E elpinoctogintillion 230 |
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!Length of these cycles |
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!Number of these cycles |
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90 nonagintillion 233 |
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|- |
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!1 |
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91 unnonagintillion 236 |
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|{0} |
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|1 |
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92 duononagintillion 239 |
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|1 |
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|- |
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93 trenonagintillion 240 |
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!2 |
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|{00}, {0E, X1, 83, 47, 29, 65} |
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94 quattuornonagintillion 243 |
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|1, 6 |
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|2 |
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95 quinnonagintillion 246 |
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|- |
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!3 |
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96 sexnonagintillion 249 |
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|{000}, {5E6} |
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|1, 1 |
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97 septennonagintillion 250 |
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|2 |
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|- |
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98 octononagintillion 253 |
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!4 |
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|{0000}, {3EE8, 8284, 6376}, {4198, 8374, 5287, 6196, 7EE4, 7375} |
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99 novemnonagintillion 256 |
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|1, 3, 6 |
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|3 |
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9X dekranonagintillion 259 |
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|- |
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!5 |
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9E elpinnonagintillion 260 |
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|{00000}, {64E66, 6EEE5}, {83E74} |
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|1, 2, 1 |
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X0 dekragintillion 263 |
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|3 |
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|- |
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X1 undekragintillion 266 |
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!6 |
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|{000000}, {420X98, X73742, 842874, 642876, 62EE86, 951963, 860X54, X40X72, X82832, 864654}, {65EE56} |
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X2 duodekragintillion 269 |
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|1, X, 1 |
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|3 |
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X3 tredekragintillion 270 |
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|- |
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!7 |
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X4 quattuordekragintillion 273 |
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|{0000000}, {841E974, X53E762, 971E943, X64E652, 960EX53, E73E741, X82E832, 984E633, 863E754}, {962E853} |
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|1, 9, 1 |
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X5 quindekragintillion 276 |
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|3 |
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|- |
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X6 sexdekragintillion 279 |
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!8 |
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|{00000000}, {4210XX98, X9737422, 87428744, 64328876, 652EE866, 961EE953, X8428732, 86528654, 6410XX76, X92EE822, 9980X323, X7646542, 8320X984, X7537642, 8430X874, X5428762, 8630X854, X540X762, X830X832, X8546632, 8520X964, X740X742, X8328832, 86546654}, {873EE744}, {X850X632} |
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X7 septendekragintillion 280 |
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|1, 20, 1, 1 |
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|4 |
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X8 octodekragintillion 283 |
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|} |
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X9 novemdekragintillion 286 |
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XX dekradekragintillion 289 |
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XE elpindekragintillion 290 |
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E0 elpagintillion 293 |
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E1 unelpagintillion 296 |
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E2 duoelpagintillion 299 |
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E3 treelpagintillion 2X0 |
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E4 quattuorelpagintillion 2X3 |
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E5 quinelpagintillion 2X6 |
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E6 sexelpagintillion 2X9 |
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E7 septenelpagintillion 2E0 |
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E8 octoelpagintillion 2E3 |
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E9 novemelpagintillion 2E6 |
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EX dekraelpagintillion 2E9 |
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EE elpinelpagintillion 300 |
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100 centillion 303 |
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(there is no consistent and widely accepted way to extend cardinals beyond centillion) |
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[[File:Dozenal digit.jpeg|500px]] |
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(numbers not coprime to 10 (dozen) only have bent lines, number coprime to 10 (dozen) only have straight lines) |
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(this is just digit notation, for the sound, k- for digits divisible by 3, t- for digits == 1 mod 3, p- for digits == 2 mod 3, -o (using Japanese sound) for digits divisible by 4, -i for digits == 1 mod 4, -u for digits == 2 mod 4, -e for digits == 3 mod 4) |
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(Japanese sound) 0 = ko, 1 = ti (just “ti”, not “chi”), 2 = pu, 3 = ke, 4 = to, 5 = pi, 6 = ku, 7 = te, 8 = po, 9 = ki, X = tu (just “tu”, not “tsu”), E = pe |
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(digit ''d'' rotating half turn is digit 10−''d'' (0 is still 0), digit ''d'' mirroring left-right is digit ''d''±6 (choose +/− to let the result number between 0 and 10) (0 is still 0, 6 is still 6), digit ''d'' mirroring up-down is digit 6−''d'' (when 0<''d''<6) or 16−''d'' (when 6<''d''<10) (0 is still 0, 6 is still 6)) |
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In this digit notation: |
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Strobogrammatic numbers: 0, 6, 1E, 2X, 39, 48, 57, 66, 75, 84, 93, X2, E1, 10E, 16E, 20X, 26X, 309, 369, 408, 468, 507, 567, 606, 666, 705, 765, 804, 864, 903, 963, X02, X62, E01, E61, 100E, 11EE, 12XE, 139E, 148E, 157E, 166E, 175E, 184E, 193E, 1X2E, 1E1E, 200X, 21EX, 22XX, 239X, 248X, 257X, 266X, 275X, 284X, 293X, 2X2X, 2E1X, 3009, 31E9, 32X9, 3399, 3489, 3579, 3669, 3759, 3849, 3939, 3X29, 3E19, 4008, 41E8, 42X8, 4398, 4488, 4578, 4668, 4758, 4848, 4938, 4X28, 4E18, 5007, 51E7, 52X7, 5397, 5487, 5577, 5667, 5757, 5847, 5937, 5X27, 5E17, 6006, 61E6, 62X6, 6396, 6486, 6576, 6666, 6756, 6846, 6936, 6X26, 6E16, 7005, 71E5, 72X5, 7395, 7485, 7575, 7665, 7755, 7845, 7935, 7X25, 7E15, 8004, 81E4, 82X4, 8394, 8484, 8574, 8664, 8754, 8844, 8934, 8X24, 8E14, 9003, 91E3, 92X3, 9393, 9483, 9573, 9663, 9753, 9843, 9933, 9X23, 9E13, X002, X1E2, X2X2, X392, X482, X572, X662, X752, X842, X932, XX22, XE12, E001, E1E1, E2X1, E391, E481, E571, E661, E751, E841, E931, EX21, EE11, ... |
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Strobogrammatic primes: 1E, 57, 75, 16E, 507, 705, E61, 148E, 157E, 184E, 1X2E, 5007, 51E7, 5397, 5487, 5X27, 5E17, 7005, 71E5, 7395, 7665, 7E15, E001, E2X1, E481, E571, E931, 1000E, 1060E, 1369E, 1468E, 1666E, 1765E, 1963E, 50007, 50607, 516E7, 526X7, 53697, 57657, 58647, 70605, 716E5, 7E615, E0001, E0601, E7651, 10570E, 10930E, 10X20E, 1139EE, 1184EE, 11X2EE, 1293XE, 12X2XE, 13489E, 13E19E, 14008E, 15007E, 152X7E, 15757E, 15847E, 16396E, 16576E, 17575E, 17665E, 17E15E, 18004E, 18664E, 18E14E, 19003E, 192X3E, 19483E, 1X482E, 1X842E, 1XE12E, 1E2X1E, 1EX21E, 500007, 501E07, 502X07, 503907, 50E107, 5148E7, 51E1E7, 5200X7, 52X2X7, 530097, 533997, 539397, 53X297, 541E87, 543987, 551E77, 557577, 55X277, 564867, 566667, 56X267, 56E167, 573957, 574857, 579357, 57X257, 580047, 582X47, 584847, 588447, 594837, 59E137, 5X1E27, 5X3927, 5E1E17, 5E6617, 701E05, 708405, 709305, 70X205, 70E105, 712XE5, 7284X5, 735795, 736695, 739395, 73X295, 744885, 74E185, 752X75, 753975, 762X65, 76E165, 775755, 782X45, 789345, 78X245, 790035, 791E35, 793935, 799335, 79E135, 7X3925, 7X5725, 7E6615, 7E8415, E05701, E09301, E0X201, E11EE1, E157E1, E184E1, E248X1, E257X1, E275X1, E30091, E33991, E35791, E40081, E44881, E50071, E51E71, E54871, E56671, E62X61, E65761, E67561, E71E51, E74851, E75751, E81E41, E98431, EX0021, EX4821, EE1E11, EE2X11, EE3911, EE8411, ... |
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Strobogrammatic squares: 0, 84, 309, 804, 87E154, ... |
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Horizontally symmetric numbers: 0, 3, 6, 9, 15, 24, 33, 42, 51, 66, 7E, 8X, 99, X8, E7, 105, 135, 165, 195, 204, 234, 264, 294, 303, 333, 363, 393, 402, 432, 462, 492, 501, 531, 561, 591, 606, 636, 666, 696, 70E, 73E, 76E, 79E, 80X, 83X, 86X, 89X, 909, 939, 969, 999, X08, X38, X68, X98, E07, E37, E67, E97, 1005, 1155, 1245, 1335, 1425, 1515, 1665, 17E5, 18X5, 1995, 1X85, 1E75, 2004, 2154, 2244, 2334, 2424, 2514, 2664, 27E4, 28X4, 2994, 2X84, 2E74, 3003, 3153, 3243, 3333, 3423, 3513, 3663, 37E3, 38X3, 3993, 3X83, 3E73, 4002, 4152, 4242, 4332, 4422, 4512, 4662, 47E2, 48X2, 4992, 4X82, 4E72, 5001, 5151, 5241, 5331, 5421, 5511, 5661, 57E1, 58X1, 5991, 5X81, 5E71, 6006, 6156, 6246, 6336, 6426, 6516, 6666, 67E6, 68X6, 6996, 6X86, 6E76, 700E, 715E, 724E, 733E, 742E, 751E, 766E, 77EE, 78XE, 799E, 7X8E, 7E7E, 800X, 815X, 824X, 833X, 842X, 851X, 866X, 87EX, 88XX, 899X, 8X8X, 8E7X, 9009, 9159, 9249, 9339, 9429, 9519, 9669, 97E9, 98X9, 9999, 9X89, 9E79, X008, X158, X248, X338, X428, X518, X668, X7E8, X8X8, X998, XX88, XE78, E007, E157, E247, E337, E427, E517, E667, E7E7, E8X7, E997, EX87, EE77, ... |
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Horizontally symmetric primes: 3, 15, 51, E7, 105, 195, 531, 591, 70E, 76E, E37, E67, E97, 1005, 1245, 1425, 1995, 5001, 5331, 5421, 57E1, 5X81, 700E, 766E, E157, E337, E8X7, EX87, 10905, 11355, 12345, 12945, 13335, 14325, 14625, 15915, 16065, 170E5, 183X5, 19395, 1E675, 50001, 50301, 54321, 54921, 56361, 573E1, 583X1, 59391, 5X681, 5E671, 7204E, 7234E, 7264E, 7303E, 7363E, 7432E, 7531E, 7666E, 7696E, 773EE, 783XE, 789XE, 7999E, 7X38E, 7X68E, 7E97E, E0007, E0907, E2047, E3037, E3637, E5617, E73E7, E83X7, E86X7, E89X7, EX387, EX687, 102405, 108X05, 114255, 118X55, 123345, 128X45, 133335, 134235, 146625, 148X25, 14X825, 152415, 155115, 156615, 161565, 163365, 16X865, 1742E5, 177EE5, 1815X5, 1866X5, 1899X5, 190095, 191595, 192495, 194295, 197E95, 1X0085, 1X5185, 1X9985, 1E0075, 1E3375, 1E6675, 1E7E75, 1E8X75, 1EX875, 502401, 506601, 50E701, 513351, 514251, 517E51, 518X51, 51X851, 522441, 52X841, 538X31, 539931, 53X831, 544221, 546621, 557E11, 55E711, 568X61, 5715E1, 578XE1, 57X8E1, 5800X1, 5815X1, 5842X1, 5851X1, 5866X1, 5899X1, 58X8X1, 594291, 5X4281, 5XE781, 5EX871, 5EE771, 70X80E, 71155E, 71245E, 717E5E, 71X85E, 71E75E, 72154E, 72514E, 72664E, 728X4E, 72E74E, 73003E, 73153E, 738X3E, 73993E, 74332E, 74E72E, 75421E, 75661E, 75E71E, 76426E, 768X6E, 7751EE, 788XXE, 79009E, 79249E, 79429E, 79X89E, 7X158E, 7X248E, 7X338E, 7XE78E, 7E517E, 7E667E, 7E8X7E, 7EE77E, E04207, E06607, E08X07, E11557, E12457, E18X57, E24247, E28X47, E34237, E35137, E39937, E40027, E44227, E46627, E54217, E62467, E68X67, E733E7, E77EE7, E78XE7, E88XX7, E8X8X7, E9X897, EX5187, EX6687, EX9987, EXX887, EE1577, EE5177, ... |
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[https://upload.wikimedia.org/wikipedia/commons/7/73/Circle_of_major_scales.svg] |
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[https://upload.wikimedia.org/wikipedia/commons/9/98/Keyboard_with_circle_of_major_scales%2C_bw.svg] |
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[https://upload.wikimedia.org/wikipedia/commons/7/75/Circle_of_fifths%2C_bw.svg] |
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[https://upload.wikimedia.org/wikipedia/commons/1/11/Keys_in_dozenal_clock.svg] |
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[https://upload.wikimedia.org/wikipedia/commons/6/6a/Dozenal_calculator_for_numbers_between_-12%5E144_and_12%5E144.jpg] |
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[https://upload.wikimedia.org/wikipedia/commons/9/96/Dozenal_12-digit_calculator.jpeg] |
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[http://dozenal.weebly.com/multiplication.html] |
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The [[self number]]s up to 600 are 1, 3, 5, 7, 9, E, 20, 31, 42, 53, 64, 75, 86, 97, X8, E9, 10X, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1X9, 1EX, 20E, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2XX, 2EE, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39X, 3XE, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48X, 49E, 4E0, 501, 512, 514, 525, 536, 547, 558, 569, 57X, 58E, 5X0, 5E1. |
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[https://puzzlewocky.com/math-fun/the-dozenal-system/] |
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The [[Friedman number]]s up to 1000 are 121=11<sup>2</sup>, 127=7×21, 135=5×31, 144=4×41, 163=3×61, 368=8<sup>6−3</sup>, 376=6×73, 441=(4+1)<sup>4</sup>, 445=5<sup>4</sup>+4. |
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[http://dozenal.weebly.com/division.html] |
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The [[Keith number]]s up to 1000 are 11, 15, 1E, 22, 2X, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, XX, EE, 125, 215, 24X, 405, 42X, 654, 80X, 8X3, X59. |
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[https://hexnet.org/content/dozenal-tau-unit-circle] |
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There are totally 71822 [[polydivisible number]]s, the largest of which is 24-digit 606890346850EX6800E036206464. However, there are no E-digit polydivisible numbers contain the digits 1 to E exactly once each. (hence there are also no 10-digit polydivisible numbers using all the digits 0 to E exactly once, since if a number with digits ''abcdefghijkl'' is a 10-digit polydivisible number using all the digits 0 to E exactly once, then {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k'', ''l''} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and then ''abcdefghijkl'' is divisible by 10, thus we have ''l'' = 0 (by [[divisibility rule]] of 10), and {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k''} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, thus a number with digits ''abcdefghijk'' is an E-digit polydivisible numbers using all the digits 1 to E exactly once). (proof: if a number with digits ''abcdefghijk'' is an E-digit polydivisible numbers using all the digits 1 to E exactly once, then {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k''} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and we have: |
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[http://gorpub.freeshell.org/dozenal/multim.html] |
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''f'' = 6 (since ''abcdef'' is divisible by 6) (by [[divisibility rule]] of 6) |
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[http://gorpub.freeshell.org/dozenal/logo_shapes_dozenal.png] |
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{''d'', ''h''} = {4, 8} (since ''abcd'' is divisible by 4 and ''abcdefgh'' is divisible by 8 (thus by 4)) (by [[divisibility rule]] of 4) |
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[http://gorpub.freeshell.org/dozenal/hammond_logo.png] |
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{''c'', ''i''} = {3, 9} (since ''abc'' is divisible by 3 and ''abcdefghi'' is divisible by 9 (thus by 3)) (by [[divisibility rule]] of 3) |
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[http://gorpub.freeshell.org/dozenal/keyb_doz.png] |
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{''b'', ''j''} = {2, X} (since ''ab'' is divisible by 2 and ''abcdefghij'' is divisible by X (thus by 2)) (by [[divisibility rule]] of 2) |
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[http://gorpub.freeshell.org/dozenal/dozclock.png] |
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thus, we have {''a'', ''e'', ''g'', ''k''} = {1, 5, 7, E} |
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[http://gorpub.freeshell.org/dozenal/printpro.png] |
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Since ''abcdefgh'' is divisible by 8, thus ''gh'' is divisible by 8 (by [[divisibility rule]] of 8), and since {''a'', ''e'', ''g'', ''k''} = {1, 5, 7, E}, thus ''g'' is odd, and ''h'' must be 4 (if ''h'' = 8 and ''g'' is odd, then ''gh'' is not divisible by 8), and since ''abcdefghi'' is divisible by 9, thus ''hi'' is divisible by 9 (by [[divisibility rule]] of 9), however, ''h'' = 4 and ''i'' is either 3 or 9, but neither 43 nor 49 is divisible by 9. |
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[http://gorpub.freeshell.org/dozenal/periodic.html] |
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If we do not require the number formed by its first 8 digits divisible by 8, then there are 2 solutions: 1X98265E347 and 7298X65E341 (neither satisfies that the number formed by its first 8 digits is divisible by 4). |
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[http://www.angelfire.com/whittenwords/measure/dozentime.htm] |
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If we do not require the number formed by its first 9 digits divisible by 9, then there are 4 solutions: 1X38E694725, 7X981634E25, 7X98E654321, and EX987634125 (only 7X98E654321 satisfies that the number formed by its first 9 digits is divisible by 3). |
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[http://dozenal.weebly.com/] |
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The candidate [[Lychrel number]]s up to 1000 are 179, 1E9, 278, 2E8, 377, 3E7, 476, 4E6, 575, 5E5, 674, 6E4, 773, 7E3, 872, 8E2, 971, 9E1, X2E, X3E, X5E, X70, XXE, XE0, E2X, E3X, E5X, EXX. The only suspected Lychrel seed numbers up to 1000 are 179, 1E9, X3E and X5E. However, it is unknown whether any Lychrel number exists. (Lychrel numbers only known to exist in these bases: E, 15, 18, 22 and all [[power of 2|powers of 2]]) |
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[http://dozenal.weebly.com/12-month-calendar.html] |
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Most numbers that end with 2 are [[nontotient]] (in fact, all nontotients < 58 except 2X end with 2), except 2 itself, the first counterexample is 92, which equals φ(X1) = φ(E<sup>2</sup>) and φ(182) = φ(2×E<sup>2</sup>), next counterexample is 362, which equals φ(381) = φ(1E<sup>2</sup>) and φ(742) = φ(2×1E<sup>2</sup>), there are only 9 such numbers ≤ 10000 (the number 2 itself is not counted), all such numbers (except the number 2 itself) are of the form φ(''p''<sup>2</sup>) = ''p''(''p''−1), where ''p'' is a prime ends with E. |
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[https://dozenal.boards.net/] |
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There is a known generalized [[Cullen prime]] for all bases ''b'' ≤ 10 (but not for next base ''b'' = 11). (no matter whether you require ''n'' ≥ ''b''−1 or not) |
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[http://www.dozenalsociety.org.uk/] |
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There is a known generalized [[Woodall prime]] for all bases ''b'' ≤ 100 (but not for next base ''b'' = 101). (no matter whether you require ''n'' ≥ ''b''−1 or not) |
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[https://docplayer.net/170950-The-dozenal-society-of-america.html] |
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There is a known generalized [[Cullen prime]] for all even bases ''b'' ≤ 100 (but not for next even base ''b'' = 102) if you don’t require ''n'' ≥ ''b''−1 |
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[https://www.mayhematics.com/j/gpj42.htm] |
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There is a known generalized [[Carol prime]] for all even bases ''b'' ≤ (100×2 + 100÷2) (=260) (but not for next even base ''b'' = 262). |
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[https://www.thegamecrafter.com/games/k6t-dozenal] |
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There is a known generalized [[Kynea prime]] for all even bases ''b'' ≤ (100×2) (=200) (but not for next even base ''b'' = 202). |
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[https://profile-stalker.to/post/B__qSK3HwZk/] |
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The generalized [[minimal prime]]s problem has at most one unsolved family for all bases ''b'' ≤ 20 (but not for ''b'' = 21). (there is one unsolved family for ''b'' = 15, 17 and 19, and there are no unsolved families for all other ''b'' ≤ 20, but for ''b'' = 21, there are 10 unsolved families) |
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[https://www.brianhetrick.com/gplsw/index.html] |
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There are no ''n''≤100 which is [[nontotient]], [[noncototient]], and [[untouchable]]. (the smallest such ''n'' is indeed the smallest even number > 100, i.e. 102) |
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[http://www.bristoljon.uk/projects/dozenal/] |
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By [[sieve of Eratosthenes]], we can cross out every composites ≤ 20 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3). (however, we cannot cross out the composite 21 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3)) |
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[https://daraghegan.com/page/dozenal-calendar.php] |
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By [[sieve of Eratosthenes]], we can cross out every composites ≤ 200 by sieve to the prime 10+1 (=11). (however, we cannot cross out the composite 201 by sieve to the prime 10+1 (=11)) |
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[http://i.imgur.com/rbP5u.jpg] |
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For all odd composites ''c'' ≤ 1000, there exists integer ''b'' coprime to ''c'' such that ''b''<sup>(''c''−1)/2</sup> ≠ ±1 (mod ''c'') (i.e. ''c'' is not [[Euler pseudoprime]] base ''b''). (this is not true for the composite ''c'' = 1001, 1001 is the smallest [[absolute Euler pseudoprime]]) |
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[http://i.imgur.com/KX6gn.jpg] |
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The [[Wagstaff number]]s <math>\frac{2^p+1}{3}</math> is prime for all odd primes ''p'' ≤ 20 (but not for ''p'' = next odd prime (25)). |
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[http://i.imgur.com/u6jke.jpg] |
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There is a known odd [[generalized Wieferich prime]] for all prime bases ''p'' ≤ 20 (but not for ''p'' = next prime (25)). |
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[https://differentbaseblocks.fandom.com/wiki/Dozenal] |
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{3}1 (i.e. 333333...3333331) cannot be prime if the number of 3’s is even, because of algebra factors (difference of two squares), besides, {3}11 (i.e. 333333...33333311) cannot be prime except the number 11 itself, because this number is divisible by 11 if the number of 3’s is even, and has algebra factors (difference of two squares) if the number of 3’s is odd. |
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[https://www.zazzle.co.uk/dozenal_tau_unit_circle_clock-256496157621674570] |
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The smallest [[Perrin pseudoprime]] is a [[near-repunit]] 111101 (note that this number is [[square]]: [[375]]<sup>2</sup>), this number only contains five 1's and one 0 (no any digit >1), and this number plus 10 is the repunit with length 6, i.e. 111111, and this number is exactly the final 6 (half of 10) digits of 123456789XE×E (a 10-digit number). |
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[https://medium.com/@Jernfrost/teaching-kids-about-alternative-number-bases-36e3cc464250] |
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The [[repunit]] prime 111 is not only the smallest [[irregular prime]] with [[irregular index]] >1, but also exactly the largest known negative [[radix|base]] −''n'' such that the [[repunit]] with length ''n'' is prime (i.e. (111^111+1)/(111+1) is prime, and 111 is the largest known such number), such number can be prime only for prime ''n'' ends with 1 or 5, except ''n'' = 3 (for positive [[radix|base]], the largest known such ''n'' is 444E, note that 444E is a near-repdigit prime, and with a repdigit (444) followed by the digit E, such number can be prime only for prime ''n'' ends with 7 or E, except ''n'' = 2 and ''n'' = 3) |
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[https://saintpat1985.wordpress.com/2016/12/31/first-blog-post/] |
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23 and 233 (but not 2333, 23333, 233333, ..., and conjectured no other such numbers) (the ratio for these two numbers is very close to 10) are both starting value set new records for number of steps to reach 1 for the Collatz (3''x''+1) sequences, and the only two numbers < 4X7 (which is also a starting value set new records for maximum value for the Collatz (3''x''+1) sequences, like the number 23) whose records of number of steps to reach 1 are ≥ (the previous record + 10). |
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[https://ideophilus.wordpress.com/2015/03/09/dozenal-numerals-with-meaningful-components/] |
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The ratio for the two smallest Weird numbers (5X and 598) is very close to 10 |
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[https://dozenaloz.wordpress.com/how-to-use-base-12/] |
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The smallest [[amicable number pair]] is (164,1E8), and the next number of them in the [[home prime]] problem are the same: 225E. (both of these two numbers get to the 9E-digit prime 51712E1X75561E754940E168440995028213825E76175409310343166811361353638X487E880858E7E8E024666872X98X3096E504E110578059E65 after 58 steps) |
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[https://www.theguardian.com/science/alexs-adventures-in-numberland/2012/dec/12/dozenalists-world-unite-tyranny-ten] |
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The “beast number” ([[666]]), when add a digit “1” before it and add another digit “1” after it, it become a [[Palindromic number|palindromic]] [[Square number|square]] 16661, it is the smallest palindromic square whose square root is not palindrome (12E), it is also the smallest palindromic square depending on [[Radix|base]]. |
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[http://www.alaricstephen.com/main-featured/2017/1/9/dozenal] |
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If we let the [[musical note]]s in an [[octave]] be numbers in the [[cyclic group]] ''Z''<sub>10</sub>: C=0, C#=1, D=2, Eb=3, E=4, F=5, F#=6, G=7, Ab=8, A=9, Bb=X, B=E (see [[pitch class]] and [[music scale]]) (thus, if we let the [[middle C]] be 0, then the notes in a [[piano]] are -33 to 40), then ''x'' and ''x''+3 are [[minor third]], ''x'' and ''x''+4 are [[major third]], ''x'' and ''x''+7 are [[perfect fifth]] (thus, we can use 7''x'' for ''x'' = 0 to E to get the [[five degree cycle]]), etc. (since an octave is 10 semitones, a minor third is 3 semitones, a major third is 4 semitones, and a perfect fifth is 7 semitones, etc.) (if we let an octave be 1, then a semitone will be 0.1, and we can write all 10 notes on a cycle, the difference of two connected notes is 26 degrees or <math>\frac{\pi}{6}</math> radians) Besides, the ''x'' major [[chord (music)|chord]] (''x'') is {''x'', ''x''+4, ''x''+7} in ''Z''<sub>10</sub>, and the ''x'' minor [[chord (music)|chord]] (''x''m) is {''x'', ''x''+3, ''x''+7} in ''Z''<sub>10</sub>, and the ''x'' [[major 7th chord]] (''x''<sup>M7</sup>) is {''x'', ''x''+4, ''x''+7, ''x''+E}, and the ''x'' [[minor 7th chord]] (''x''<sup>m7</sup>) is {''x'', ''x''+3, ''x''+7, ''x''+X}, and the ''x'' [[dominant 7th chord]] (''x''<sup>7</sup>) is {''x'', ''x''+4, ''x''+7, ''x''+X}, and the ''x'' [[diminished triad|diminished 7th triad]] (''x''<sup>dim7</sup>) is {''x'', ''x''+3, ''x''+6, ''x''+9}, since the frequency of ''x'' and ''x''+6 is not simple integer fraction, they are not [[harmonic]], and this diminished 7th triad is corresponding the [[beast number]] 666 (three 6's) (also, ''x'' and ''x''+6 are [[tritone]], which is not harmonic). Besides, ''x'' [[major scale]] uses the notes {''x'', ''x''+2, ''x''+4, ''x''+5, ''x''+7, ''x''+9, ''x''+E}, and ''x'' [[minor scale]] uses the notes {''x'', ''x''+2, ''x''+3, ''x''+5, ''x''+7, ''x''+8, ''x''+X}. Besides, the [[frequency]] of ''x''+10 is twice as that of ''x'', the frequency of ''x''+7 is 1.6 (=3/2) times as that of ''x'', and the frequency of ''x''+5 is 1.4 (=4/3) times as that of ''x'', they are all simple integer fractions (ratios of small integers), and they all have at most one digit after the duodecimal point, and we can found that 1.6<sup>10</sup> = X9.8E5809 is very close to 2<sup>7</sup> = X8, since 2<sup>17</sup> = 2134X8 is very close to 3<sup>10</sup> = 217669, the simple frequency fractions found for the scales are only 0.6, 0.8, 0.9, 1.4, 1.6 and 2, however, since the frequency of ''x''+10 is twice as that of ''x'', thus the frequency of ''x''+1 (i.e. a [[semitone]] higher than ''x'') is [[twelfth root of two|<math>\sqrt[10]{2}</math>]] (=2<sup>0.1</sup>) times as that of ''x''. Let ''f''(''x'') be the frequency of ''x'', then we have ''f''(2)/''f''(0) = 9/8 (=1.16), ''f''(4)/''f''(2) = X/9 (=1.14), and ''f''(5)/''f''(4) = 14/13 (this number is very close to <math>\sqrt[10]{2}</math>), and thus we have that ''f''(5)/''f''(0) = (9/8) × (X/9) × (14/13) = 4/3. Also, we can found that 2<sup>0.5</sup> is very close to 1.4, and 2<sup>0.7</sup> is very close to 1.6. |
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[https://www.reddit.com/r/dozenal/comments/cv7z72/a_little_numeral_i_made/] |
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All orders of non-[[cyclic group|cyclic]] [[simple group]] end with 0 (thus, all orders of [[solvable group|unsolvable group]] end with 0), however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group (thus, all groups with order < 50 are solvable), (50 is the order of the [[alternating group]] ''A''<sub>5</sub>, which is a non-cyclic simple group, and thus an unsolvable group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not completely true (although this is true for all orders ≤ 14000), the smallest counterexample is 14X28, however, all such orders are divisible by 4 and either 3 or 5 (i.e. divisible by either 10 or 18), and all such orders have at least 3 distinct prime factors, by these conditions, the smallest possible such order is indeed 50 = 2<sup>2</sup> × 3 × 5, next possible such order is 70 = 2<sup>2</sup> × 3 × 7, however, by [[Sylow theorems]], the number of Sylow 7-subgroups of all groups with order 70 (i.e. the number of subgroups with order 7 of all groups with order 70) is congruent to 1 mod 7 and divides 70, hence must be 1, thus the subgroup with order 7 is a [[normal subgroup]] of the group with order 70, thus all groups with order 70 have a nontrivial normal subgroup and cannot be simple groups) |
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[https://www.pinterest.com/pin/50806302027129652/] |
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The [[probability]] for rolling a 6 on a [[dice]] is 0.2 or 20%, and the probability for rolling at least one 6 on a dice in 3 rolls is 0.508 (less than one half or 60%), and the probability for rolling at least one 6 on a dice in 4 rolls is 0.6268 (more than one half or 60%), and the probability for rolling a "double 6" on two dices is 0.04 or 4%, and the probability for rolling at least one "double 6" on two dices in 20 rolls is 0.5X9190... (less than one half or 60%), and the probability for rolling at least one "double 6" on two dices in 21 rolls is 0.609685... (more than one half or 60%). |
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[https://allthingsdozenal.blogspot.com/2016/08/dozenal-calculator-20-is-here.html] |
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<span style="color:blue;">0</span>, <span style="color:green;">1</span>, <span style="color:red;">2</span>, <span style="color:green;">3</span>, <span style="color:red;">4</span>, <span style="color:green;">5</span>, <span style="color:green;">6</span>, <span style="color:red;">7</span>, <span style="color:green;">8</span>, <span style="color:red;">9</span>, <span style="color:green;">X</span>, <span style="color:blue;">E</span> |
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[http://www.dozenalsociety.org.uk/pdfs/Dozenite.pdf] |
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<span style="color:red;">appear in the repeating digits of 1/5 (exactly the even digits ≤ 5 (except 0 (the smallest digit)) and the odd digits ≥ 6 (except E (the largest digit)))</span> |
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[http://22nina73g.changeip.com/Dozenal-watch.html] |
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<span style="color:green;">appear in the repeating digits of 1/7 (exactly the odd digits ≤ 5 and the even digits ≥ 6)</span> |
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[https://www.reddit.com/r/dozenal/comments/4pmaxx/proposed_dozenal_numerals/] |
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<span style="color:blue;">appear in the repeating digits of 1/11 (exactly the smallest digit (0) and the largest digit (E))</span> |
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[https://www.reddit.com/r/dozenal/comments/7h7ve6/just_watched_numberphile_on_base_12_my_versions/] |
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(note that the number of digits ≤5 is equal to the number of digits ≥6 (both are 6, which is equal to half of the [[radix|base]] we used in this wiki (10)), and all digits are either ≤5 or ≥6, but not both) |
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[http://gorpub.freeshell.org/dozenal/newdigs.html] |
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(note that all of 5, 7, and 11 are primes, and they are the only three primes ≤ 10+1 (=11) and divides neither 10 nor 10−1 (=E)) |
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[https://www.deviantart.com/treisaran/art/Dozenal-Numeral-Sets-Comparison-319565301] |
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253 = 3<sup>3</sup> × 11 = 3<sup>3</sup> × (10+1) |
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[https://www.deviantart.com/treisaran/art/Standard-Dozenal-Primes-Table-343133864] |
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254 = 2<sup>5</sup> × E = 2<sup>5</sup> × (10−1) |
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1253 = 3<sup>3</sup> × 7 × E = 3<sup>3</sup> × (6+1) × (10−1) |
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[http://edspi31415.blogspot.com/2017/01/base-12-arithmetic-dozenal-society.html] |
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1254 = 2<sup>5</sup> × 5 × 11 = 2<sup>5</sup> × (6−1) × (10+1) |
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253<sup>−1</sup> = 0.004<span style="text-decoration: overline;">E0</span> |
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[https://www.tapatalk.com/groups/dozensonline/imagine-the-digital-world-fully-dozenal-t1306.html] |
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254<sup>−1</sup> = 0.004<span style="text-decoration: overline;">X</span> |
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(Note that 6 is half of 10, and 2 and 3 are the only prime factors of 10 (also the only prime factors of 6), and 2<sup>5</sup> and 3<sup>3</sup> are the smallest power of 2 and 3 which does not divide 100, besides, 2, 3, 6−1, 6+1, 10−1, 10+1 are exactly the first primes) |
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[https://www.scienceabc.com/eyeopeners/why-we-should-already-use-base-12-instead-of-base-10.html] |
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All ''k'' ≤ 394 has at least one prime <math>\Phi_k(n)</math> with 2 ≤ ''n'' ≤ 600, but.... |
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[https://upload.wikimedia.org/wikipedia/commons/f/ff/Circle_of_fifths_unrolled%2C_dozenal.svg] |
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395 = 5 × 91 (5 times a prime number), <math>\Phi_{395}(n)</math> isn't prime until ''n'' = 1696, however, the smallest ''n'' ≥ 2 such that <math>\Phi_{395}(-n)</math> is prime is only 10 (10 is exactly the [[radix|base]] of this wiki). |
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[https://www.reddit.com/r/dozenal/comments/20j2ck/dozenal_approximation_of_pi/] |
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3X1 = 7 × 67 (7 times a prime number), <math>\Phi_{3\mathcal{X}1}(n)</math> isn't prime until ''n'' = 989, however, the smallest ''n'' ≥ 2 such that <math>\Phi_{395}(-n)</math> is prime is only 2 (2 is exactly the smallest possible [[radix|base]]). |
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[http://www.stevenjacks.com/apps/dozenal-calculator/] |
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(Note that 395 is the only divisor ''d'' of 76X such that <math>\Phi_d(10)</math> is not prime, 76X has 8 divisors, and for all divisors ''d'' except 395, <math>\Phi_d(10)</math> is prime, thus the number 10^76X−1 is product of 7 unique primes (primes ''p'' which there is no other prime ''q'' such that the period length of the dozenal expansion of its reciprocal, 1/''p'', is equal to the period length of the reciprocal of ''q'', 1/''q'') and the composite number <math>\Phi_{395}(10)</math>, and 76X is conjectured to be the largest number such that for all divisors ''d'' except one number, <math>\Phi_d(10)</math> is prime (note that 10 is exactly the [[radix|base]] of this wiki, and if we change the number 10 to other small numbers (≤20), then there is no number as large as 76X such that for all divisors ''d'' except one number, <math>\Phi_d(b)</math> is prime), besides, 76X+1 (=76E) is prime, by Fermat's little theorem (every prime ''p'' not dividing ''b'' must divide <math>\Phi_d(b)</math> for some ''d'' dividing ''p''−1), since 76E > 10, 76E does not divide 10, thus 76E must divide <math>\Phi_d(10)</math> for some ''d'' dividing 76X, however, since <math>\Phi_d(10)</math> for all ''d'' dividing 76X except 395 is prime (and not equal to 76E), 76E can only divide <math>\Phi_{395}(10)</math>, and the period length for 1/76E must be 395) |
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[http://www.stevenjacks.com/learn/dozenal/time-keeping/how-to/] |
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Besides, the primes 5 and 7 are |
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[https://medium.com/@n.patti/using-the-duodecimal-system-will-make-your-day-to-day-life-much-easier-9cf174109ddb] |
Revision as of 03:37, 7 September 2020
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −0.1 (which equals the negative value of the reciprocal of 10).
Both 10±1 are primes, both (10±2)/2 are primes, both (10±3)/3 are primes, both 10±4 are powers of 2 (also common bases for computer, base 8 and base 14), both 10±5 (also 10±7 and 10±E) are primes, and both 10±6 are regular numbers (i.e. 3-smooth numbers) (i.e. numbers n such that 1/n terminates in dozenal) (also twice a power of 3).
Both 100±1 are semiprimes, both (100±2)/2 are primes, (100−d)/d is prime for all divisors d of 10 except 1 and 4 (note that this number for d = 1 and 4 cannot be primes, because of algebra factors: 100−1 = (10−1) × (10+1), and 100−4 = (10−2) × (10+2), and can only be semiprimes), both 100±5 are primes, both 100±7 are primes, both (100±8)/8 are primes, both 100±E are semiprimes, both (100±10)/10 are primes (Note: (100−3)/3 and (100−6)/6 are primes, but (100+3)/3 and (100+6)/6 are prime squares, also, (100+9)/9 is prime, but (100−9)/9 also has algebra factors (100−9 = (10−3) × (10+3)) and cannot be prime)
All squares end with square digits (i.e. end with 0, 1, 4 or 9), if n is divisible by both 2 and 3, then n2 ends with 0, if n is not divisible by 2 or 3, then n2 ends with 1, if n is divisible by 2 but not by 3, then n2 ends with 4, if n is not divisible by 2 but by 3, then n2 ends with 9. If the unit digit of n2 is 0, then the dozens digit of n2 is either 0 or 3, if the unit digit of n2 is 1, then the dozens digit of n2 is even, if the unit digit of n2 is 4, then the dozen digit of n2 is 0, 1, 4, 5, 8 or 9, if the unit digit of n2 is 9, then the dozen digit of n2 is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1)
The numbers n such that the concatenation of n and the unit (1), i.e. 10n+1 (all squares of primes except 4 and 9 are of this form), is square, are all even numbers, and the half of these n are exactly the generalized pentagonal numbers, and such numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem (the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
In other words,
The exponents 1, 2, 5, 7, 10, 13, 1X, 22, ... on the right hand side are given by the formula gk = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, −3, 4, −4, ... and are called (generalized) pentagonal numbers. This holds as an identity of convergent power series for , and also as an identity of formal power series.
A striking feature of this formula is the amount of cancellation in the expansion of the product), also, the identity implies a marvelous recurrence for calculating , the number of partitions of n (p(n)):
or more formally,
Also the sum of divisors of n (σ(n)):
but if the last term is σ(0) (this situation appears if and only if n itself is generalized pentagonal number, i.e. the concatenation of 2n and 1 is square), then we change it to n.
where the summation is over all nonzero integers k (positive and negative) and is the kth generalized pentagonal number. Since for all , the series will eventually become zeroes, enabling discrete calculation, besides, generalized pentagonal numbers are closely related to centered hexagonal numbers (also called hex numbers, the hex numbers are end with 1, 7, 7, 1, 1, 7, 7, 1, ...). When the array corresponding to a centered hexagonal number is divided between its middle row and an adjacent row, it appears as the sum of two generalized pentagonal numbers, with the larger piece being a pentagonal number proper.
The digital root of a square is 1, 3, 4, 5 or E.
No repdigits with more than one digit are squares, in fact, a square cannot end with three same digits except 000.
No four-digit palindromic numbers are squares. (we can easily to prove it, since all four-digit palindromic number are divisible by 11, and since they are squares, thus they must be divisible by 112 = 121, and the only four-digit palindromic number divisible by 121 are 1331, 2662, 3993, 5225, 6556, 7887, 8EE8, 9119, X44X and E77E, but none of them are squares)
n | n-digit palindromic squares | square roots | number of n-digit palindromic squares |
---|---|---|---|
1 | 1, 4, 9 | 1, 2, 3 | 3 |
2 | none | none | 0 |
3 | 121, 484 | 11, 22 | 2 |
4 | none | none | 0 |
5 | 10201, 12321, 14641, 16661, 16E61, 40804, 41414, 44944 | 101, 111, 121, 12E, 131, 202, 204, 212 | 8 |
6 | 160061 | 42E | 1 |
7 | 1002001, 102X201, 1093901, 1234321, 148X841, 4008004, 445X544, 49XXX94 | 1001, 1015, 1047, 1111, 1221, 2002, 2112, 2244 | 8 |
8 | none | none | 0 |
9 | 100020001, 102030201, 104060401, 1060E0601, 121242121, 123454321, 125686521, 1420E0241, 1444X4441, 1468E8641, 14X797X41, 1621E1261, 163151361, 1XX222XX1, 400080004, 404090404, 410212014, 4414X4144, 4456E6544, 496787694, 963848369 | 10001, 10101, 10201, 10301, 11011, 11111, 11211, 11E21, 12021, 12121, 1229E, 1292E, 12977, 14685, 20002, 20102, 20304, 21012, 21112, 22344, 31053 | 19 |
X | 1642662461 | 434X5 | 1 |
E | 10000200001, 10221412201, 10444X44401, 12102420121, 12345654321, 141E1E1E141, 14404X40441, 16497679461, 40000800004, 40441X14404, 41496869414, 44104X40144, 49635653694 | 100001, 101101, 102201, 110011, 111111, 11E13E, 120021, 12X391, 200002, 201102, 204204, 210012, 223344 | 11 |
10 | none | none | 0 |
It is conjectured that if n is divisible by 4, then there are no n-digit palindromic squares.
Rn2 (where Rn is the repunit with length n) is a palindromic number for n ≤ E, but not for n ≥ 10 (thus, for all odd number n ≤ 19, there is n-digit palindromic square 123...321), besides, 11n (also 1{0}1n, i.e. 101n, 1001n, 10001n, etc.) is a palindromic number for n ≤ 5, but not for n ≥ 6, and it is conjectured that no palindromic numbers are n-th powers if n ≥ 6.
The square numbers using no more than two distinct digits are 0, 1, 4, 9, 14, 21, 30, 41, 54, 69, 84, X1, 100, 121, 144, 344, 400, 441, 484, 554, 900, 3000, 4344, 9944, 10000, 11XX1, 16661, 40000, 41414, 44944, 47744, 66969, 90000, 111101, 114144, 300000, 444404, 454554, 999909, 1000000, 1141144, 3333030, 4000000, 4544554, 9000000, 11110100, 30000000, 41144144, 44440400, 99990900, XXXXXXX1, 100000000, 333303000, 400000000, 900000000, 1111010000, 3000000000, 4444040000, 9999090000, 10000000000, 33330300000, 40000000000, 90000000000, 111101000000, 300000000000, 444404000000, 999909000000, 1000000000000, ...
41414 is the largest undulating square (of the form ababab...), note that the only 2 distinct digits in it (1 and 4) and the only 2 distinct two-digit numbers formed by its digits (14 and 41) are also all squares.
A cube can end with all digits except 2, 6 and X (in fact, no perfect powers end with 2, 6 or X), if n is not congruent to 2 mod 4, then n3 ends with the same digit as n; if n is congruent to 2 mod 4, then n3 ends with the digit (the last digit of n +− 6).
The cube numbers using no more than two distinct digits are 0, 1, 8, 23, 54, X5, 1000, 1331, 8000, 1000000, 8000000, 1000000000, 8000000000, 1000000000000, ...
The digital root of a cube can be any number.
21 and 201 are both squares, and it is conjectured that no other numbers of the form 2000...0001 (i.e. of the form 2×10n+1) are squares.
10814 and 100814 are both squares, and 10854 and 100853 are both cubes.
If k≥2, then nk+2 ends with the same digit as nk, thus, if i≥2, j≥2 and i and j have the same parity, then ni and nj end with the same digit.
If x^2 + y^2 = z^2 (that is, {x, y, z} is a Pythagorean triple, then xy end with 0 (and thus xyz also end with 0).
Squares (and every powers) of 0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, E3854, 1E3854, X08369, ... end with the same digits as the number itself. (since they are automorphic numbers, from the only four solutions of x2−x=0 in the ring of 10-adic numbers (dozadic numbers), these solutions are 0, 1, ...2E21E61E3854 and ...909X05X08369, since 10 is neither a prime nor a prime power, the ring of the 10-adic numbers is not a field, thus there are solutions other than 0 and 1 for this equation in 10-adic numbers)
The triangular numbers using no more than two distinct digits are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, 77, 89, X0, E4, 191, 303, 446, 550, 633, 66X, 6X6, 1117, 3X3X, 3EE3, 6060, 6161, 6366, 6999, 6EE6, 8989, 9779, 23223, 35553, 50050, 77677, 113113, 303333, 331331, 600600, X33X33, 3030330, 60006000, 333666333, 6000060000, 600000600000, ...
The pronic numbers using no more than two distinct digits are 0, 2, 6, 10, 18, 26, 36, 48, 60, 76, 92, E0, 110, 606, 656, 992, XX0, EE6, 1118, 2232, 7878, EE00, 10100, 33330, 46446, 6XXX6, X00X0, 118118, 226226, 606666, 662662, EEE000, 1001000, 6060660, EEEE0000, 100010000, EEEEE00000, 10000100000, EEEEEE000000, 1000001000000, ...
Except for 6 and 24, all even perfect numbers end with 54. Additionally, except for 6, 24 and 354, all even perfect numbers end with 054 or 854. Besides, if any odd perfect number exists, then it must end with 1, 09, 39, 69 or 99.
The digital root of an even perfect number is 1, 4, 6 or X.
Since 10 is the smallest abundant number, all numbers end with 0 are abundant numbers, besides, all numbers end with 6 except 6 itself are also abundant numbers.
k n |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 20 | Period |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 2 |
3 | 1 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 2 |
4 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 2 |
6 | 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 2 |
8 | 1 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 2 |
9 | 1 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 1 |
X | 1 | X | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | 2 |
10 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
12 | 1 | 2 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 2 |
13 | 1 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 3 | 9 | 2 |
14 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
15 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 5 | 1 | 2 |
16 | 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
17 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 7 | 1 | 2 |
18 | 1 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 2 |
19 | 1 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 1 |
1X | 1 | X | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 |
1E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | E | 1 | 2 |
20 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
The period of the unit digits of powers of a number must be a divisor of 2 (= λ(10), where λ is the Carmichael function).
n | possible unit digit of an nth power |
---|---|
0 | 1 |
1 | any number |
even number ≥ 2 | 0, 1, 4, 9 (the square digits) |
odd number ≥ 3 | 0, 1, 3, 4, 5, 7, 8, 9, E (all digits != 2 mod 4) |
k n |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 20 | Period |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 01 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 1 |
1 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 01 | 1 |
2 | 01 | 02 | 04 | 08 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | 6 |
3 | 01 | 03 | 09 | 23 | 69 | 83 | 09 | 23 | 69 | 83 | 09 | 23 | 69 | 83 | 09 | 23 | 69 | 83 | 09 | 23 | 69 | 83 | 09 | 23 | 69 | 4 |
4 | 01 | 04 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 3 |
5 | 01 | 05 | 21 | X5 | 41 | 85 | 61 | 65 | 81 | 45 | X1 | 25 | 01 | 05 | 21 | X5 | 41 | 85 | 61 | 65 | 81 | 45 | X1 | 25 | 01 | 10 |
6 | 01 | 06 | 30 | 60 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 1 |
7 | 01 | 07 | 41 | 47 | 81 | 87 | 01 | 07 | 41 | 47 | 81 | 87 | 01 | 07 | 41 | 47 | 81 | 87 | 01 | 07 | 41 | 47 | 81 | 87 | 01 | 6 |
8 | 01 | 08 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 68 | 54 | 2 |
9 | 01 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 09 | 69 | 2 |
X | 01 | 0X | 84 | E4 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 54 | 1 |
E | 01 | 0E | X1 | 2E | 81 | 4E | 61 | 6E | 41 | 8E | 21 | XE | 01 | 0E | X1 | 2E | 81 | 4E | 61 | 6E | 41 | 8E | 21 | XE | 01 | 10 |
10 | 01 | 10 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 1 |
11 | 01 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 | 91 | X1 | E1 | 01 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 | 91 | X1 | E1 | 01 | 10 |
12 | 01 | 12 | 44 | 08 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 54 | 28 | 14 | 68 | 94 | X8 | 54 | 6 |
13 | 01 | 13 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 53 | 69 | 2 |
14 | 01 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 14 | 94 | 54 | 3 |
15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 15 | 01 | 2 |
16 | 01 | 16 | 30 | 60 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 1 |
17 | 01 | 17 | 61 | 77 | 01 | 17 | 61 | 77 | 01 | 17 | 61 | 77 | 01 | 17 | 61 | 77 | 01 | 17 | 61 | 77 | 01 | 17 | 61 | 77 | 01 | 4 |
18 | 01 | 18 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | X8 | 94 | 68 | 14 | 28 | 54 | 6 |
19 | 01 | 19 | 09 | 39 | 69 | 99 | 09 | 39 | 69 | 99 | 09 | 39 | 69 | 99 | 09 | 39 | 69 | 99 | 09 | 39 | 69 | 99 | 09 | 39 | 69 | 4 |
1X | 01 | 1X | 44 | E4 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 94 | 14 | 54 | 3 |
1E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 1E | 81 | 5E | 41 | 9E | 01 | 6 |
20 | 01 | 20 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 1 |
The period of the final two digits of powers of a number must be a divisor of 10 (= λ(100)).
More generally, for every n≥2, the period of the final n digits of powers of a number must be a divisor of 10n−1 (= λ(10n)).
k n |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 1X | 1E | 20 | Period |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
3 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 5 |
4 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 5 |
5 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 5 |
6 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X |
7 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X |
8 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X |
9 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 5 |
X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | 2 |
E | 1 | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | 1 |
10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
11 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
12 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 1 | 3 | 9 | 5 | 4 | 5 |
13 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 1 | 4 | 5 | 9 | 3 | 5 |
14 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 1 | 5 | 3 | 4 | 9 | 5 |
15 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X | 5 | 8 | 4 | 2 | 1 | 6 | 3 | 7 | 9 | X |
16 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X | 4 | 6 | 9 | 8 | 1 | 7 | 5 | 2 | 3 | X |
17 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X | 3 | 2 | 5 | 7 | 1 | 8 | 9 | 6 | 4 | X |
18 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 1 | 9 | 4 | 3 | 5 | 5 |
19 | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | X | 1 | 2 |
1X | 1 | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | E | 1 |
1E | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
20 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X | 9 | 7 | 3 | 6 | 1 | 2 | 4 | 8 | 5 | X |
The period of the digital roots of powers of a number must be a divisor of X (= λ(E)).
n | possible digital root of an nth power |
---|---|
0 | 1 |
= 1, 3, 7, 9 (mod X) | any number |
= 2, 4, 6, 8 (mod X) | 1, 3, 4, 5, 9, E |
= 5 (mod X) | 1, X, E |
> 0 and divisible by X | 1, E |
The unit digit of a Fibonacci number can be any digit except 6 (if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a Lucas number cannot be 0 or 9 (thus, no Lucas number is divisible by 10), besides, if a Lucas number ends with 2, then it must end with 0002, i.e., this number is congruent to 2 mod 104.
In the following table, Fn is the n-th Fibonacci number, and Ln is the n-th Lucas number.
n | Fn | digit root of Fn | Ln | digit root of Ln | n | Fn | digit root of Fn | Ln | digit root of Ln |
---|---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 21 | 37501 | 5 | 81101 | E |
2 | 1 | 1 | 3 | 3 | 22 | 5X301 | 8 | 111103 | 7 |
3 | 2 | 2 | 4 | 4 | 23 | 95802 | 2 | 192204 | 7 |
4 | 3 | 3 | 7 | 7 | 24 | 133E03 | X | 2X3307 | 3 |
5 | 5 | 5 | E | E | 25 | 209705 | 1 | 47550E | X |
6 | 8 | 8 | 16 | 7 | 26 | 341608 | E | 758816 | 2 |
7 | 11 | 2 | 25 | 7 | 27 | 54E111 | 1 | 1012125 | 1 |
8 | 19 | X | 3E | 3 | 28 | 890719 | 1 | 176X93E | 3 |
9 | 2X | 1 | 64 | X | 29 | 121E82X | 2 | 2780X64 | 4 |
X | 47 | E | X3 | 2 | 2X | 1XE0347 | 3 | 432E7X3 | 7 |
E | 75 | 1 | 147 | 1 | 2E | 310EE75 | 5 | 6XE0647 | E |
10 | 100 | 1 | 22X | 3 | 30 | 5000300 | 8 | E22022X | 7 |
11 | 175 | 2 | 375 | 4 | 31 | 8110275 | 2 | 16110875 | 7 |
12 | 275 | 3 | 5X3 | 7 | 32 | 11110575 | X | 25330XX3 | 3 |
13 | 42X | 5 | 958 | E | 33 | 1922082X | 1 | 3E441758 | X |
14 | 6X3 | 8 | 133E | 7 | 34 | 2X3311X3 | E | 6477263E | 2 |
15 | E11 | 2 | 2097 | 7 | 35 | 47551X11 | 1 | X3EE4197 | 1 |
16 | 15E4 | X | 3416 | 3 | 36 | 75882EE4 | 1 | 148766816 | 3 |
17 | 2505 | 1 | 54E1 | X | 37 | 101214X05 | 2 | 23075X9E1 | 4 |
18 | 3XE9 | E | 8907 | 2 | 38 | 176X979E9 | 3 | 379305607 | 7 |
19 | 6402 | 1 | 121E8 | 1 | 39 | 2780E0802 | 5 | 5X9X643E8 | E |
1X | X2EE | 1 | 1XE03 | 3 | 3X | 432E885EE | 8 | 967169X03 | 7 |
1E | 14701 | 2 | 310EE | 4 | 3E | 6XE079201 | 2 | 13550121EE | 7 |
20 | 22X00 | 3 | 50002 | 7 | 40 | E22045800 | X | 2100180002 | 3 |
(Note that F2X begins with L1X, and F2E begins with L1E)
The period of the digit root of Fibonacci numbers is X.
The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final n digits is 2×10n−1 (n ≥ 2). (see Pisano period)
There are only 13 possible values (of the totally 100 values, thus only 13%) of the final two digits of a Fibonacci number (see OEIS: A066853).
Except 0 = F0 and 1 = F1 = F2, the only square Fibonacci number is 100 = F10 (100 is exactly the square of 10), thus, 10 is the only base such that 100 is a Fibonacci number (since 100 in a base is just the square of this base, and 0 and 1 cannot be the base of numeral system), and thus we can make the near value of the golden ratio: F11/F10 = 175/100 = 1.75 (since the ratio of two connected Fibonacci numbers is close to the golden ratio, as the numbers get large). Besides, the only cube Fibonacci number is 8 = F6.
The recurring dozenal of 1/F(n) terminates if and only if n is a divisor of 10. (10 is exactly the base of dozenal)
Factors of the Fibonacci numbers:
F(10) = 100 = 102
(F(10) is exactly the square of 10, and F(10) is also the largest Fibonacci number which is square)
F(10−1) = F(E) = 75 F(10+1) = F(11) = 175
75 × 175 = 10001 = 104 + 1
(10±1 are both primes, and F(10±1) are also both primes) (104 + 1 = 10001 is a semiprime, and 104 − 1 is a member of betrothed number pair, together with 5600)
F(10−2) = F(X) = 47 = 5 × E F(10+2) = F(12) = 275 = 25 × 11
5 × 25 = 101 = 102 + 1 E × 11 = EE = 102 − 1
(10±2 are both semiprimes, and F(10±2) are also both semiprimes) (102 ± 1 are both semiprimes, and 5, E, 25, and 11 are all primes)
n | 2n | n | 2n | n | 2n | n | 2n | n | 2n | n | 2n |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 21 | E2X20X8 | 41 | 5317E5804588X8 | 61 | 256906X1X93096E8934X8 | 81 | 11X12X743504482569888538X0X8 | X1 | 65933E8691303X448E712227X7E11448X8 |
2 | 4 | 22 | 1X584194 | 42 | X633XE408E5594 | 62 | 4E161183966171E566994 | 82 | 238259286X08944E17554X758194 | X2 | 10E667E51626078895E22445393X2289594 |
3 | 8 | 23 | 38E48368 | 43 | 190679X815XXE68 | 63 | 9X302347710323XE11768 | 83 | 4744E6551815689X32XX992E4368 | X3 | 21E113XX305013556EX4488X76784556E68 |
4 | 14 | 24 | 75X94714 | 44 | 361137942E99E14 | 64 | 1786046932206479X23314 | 84 | 9289E0XX342E15786599765X8714 | X4 | 43X2279860X026XE1E88955931348XE1E14 |
5 | 28 | 25 | 12E969228 | 45 | 702273685E77X28 | 65 | 3350091664410937846628 | 85 | 16557X198685X2E350E7730E95228 | X5 | 878453750180519X3E556XE6626959X3X28 |
6 | 54 | 26 | 25E716454 | 46 | 120452714EE33854 | 66 | 66X0163108821673491054 | 86 | 30XE3837514E85X6X1E3261E6X454 | X6 | 15348X72X0340X3787XXE19E10516E787854 |
7 | X8 | 27 | 4EE2308X8 | 47 | 2408X5229EX674X8 | 67 | 111803062154431269620X8 | 87 | 619X7472X29E4E9183X6503E188X8 | X7 | 2X695925806818735399X37X20X31E3534X8 |
8 | 194 | 28 | 9EX461594 | 48 | 48158X457E912994 | 68 | 2234061042X886251704194 | 88 | 103792925857X9E634790X07X35594 | X8 | 5916E64E41143526X777873841863X6X6994 |
9 | 368 | 29 | 17E8902E68 | 49 | 942E588E3E625768 | 69 | 446810208595504X3208368 | 89 | 20736564E4E397E06936181386XE68 | X9 | E631E09X82286X5193335274835079191768 |
X | 714 | 2X | 33E5605E14 | 4X | 1685XE55X7E04E314 | 6X | 891420414E6XX0986414714 | 8X | 41270E09X9X773X116703427519E14 | XX | 1E063X179445518X36666X52946X136363314 |
E | 1228 | 2E | 67XE00EX28 | 4E | 314E9XXE93X09X628 | 6E | 1562840829E1981750829228 | 8E | 82521X179793278231206852X37X28 | XE | 3X107833688XX358711118X56918270706628 |
10 | 2454 | 30 | 1139X01E854 | 50 | 629E799E678179054 | 70 | 2E05481457X37432X1456454 | 90 | 144X4383373665344624114X5873854 | E0 | 78213467155986E52222358E1634521211054 |
11 | 48X8 | 31 | 2277803E4X8 | 51 | 1057E377E1343360X8 | 71 | 5X0X9428E3872865828E08X8 | 91 | 2898874672710X6890482298E5274X8 | E1 | 1344269122XE751XX44446E5X3068X424220X8 |
12 | 9594 | 32 | 4533407X994 | 52 | 20E3X733X268670194 | 72 | E8196855X752550E455X1594 | 92 | 557552912522191560944575XX52994 | E2 | 26885162459E2X39888891XE86115884844194 |
13 | 16E68 | 33 | 8X668139768 | 53 | 41X792678515120368 | 73 | 1E43714XE92X4XX1X8XE82E68 | 93 | XE2XX5624X44362E01688E2E98X5768 | E3 | 5154X3048E7X58775555639E5022E549488368 |
14 | 31E14 | 34 | 159114277314 | 54 | 839365134X2X240714 | 74 | 3X872299E6589983959E45E14 | 94 | 19X598E049888705X03155X5E758E314 | E4 | X2X986095E38E532XXXE077XX045XX96954714 |
15 | 63X28 | 35 | 2E6228532628 | 55 | 147670X269858481228 | 75 | 79524577E0E577476E7X8EX28 | 95 | 378E75X09755520E8062XE8EE2E5X628 | E5 | 185975016EX75XX65999X1339808E99716X9228 |
16 | 107854 | 36 | 5E0454X65054 | 56 | 29312185174E4942454 | 76 | 136X48E33X1XE32931E395E854 | 96 | 735E2E8172XXX41E41059E5EX5XE9054 | E6 | 34E72X031E92E990E77782677415E7723196454 |
17 | 2134X8 | 37 | EX08X990X0X8 | 57 | 5662434X329X96848X8 | 77 | 271895X67839X65663X76EE4X8 | 97 | 126EX5E432599883X820E7XEE8E9E60X8 | E7 | 69E258063E65E761E3334513282EE32463708X8 |
18 | 426994 | 38 | 1E81597618194 | 58 | E104869865797149594 | 78 | 52356E91347790E107931EX994 | 98 | 251E8EX864E775479441E39EE5E7E0194 | E8 | 117X4E4107E0EE303X6668X26545EX6490721594 |
19 | 851768 | 39 | 3E42E73034368 | 59 | 1X20951750E372296E68 | 79 | X46E1E62693361X213663E9768 | 99 | 4X3E5E9509E32X936883X77EXEE3X0368 | E9 | 23389X8213X1EX60791115850X8EE90961242E68 |
1X | 14X3314 | 3X | 7X85E26068714 | 5X | 38416X32X1X724571E14 | 7X | 1891X3E051667038427107E7314 | 9X | 987XEE6X17X659671547933E9EX780714 | EX | 467579442783E90136222E4X195EE61702485E14 |
1E | 2986628 | 3E | 1394EX50115228 | 5E | 74831865839248E23X28 | 7E | 356387X0X3112074852213E2628 | 9E | 17539EE183390E7122X93667E7E9341228 | EE | 912E36885347E60270445X9836EEE0320494EX28 |
20 | 5751054 | 40 | 2769E8X022X454 | 60 | 12946350E476495X47854 | 80 | 6E075381862241294X4427X5054 | X0 | 32X77EX346761E2245967113E3E6682454 | 100 | 1625X7154X693E0052088E97471EEX0640969E854 |
For all digits 1 ≤ d ≤ X (i.e. all digits other than the largest digit (E)), there exists 0 ≤ n ≤ 20 such that 2n starts with the digit d. (This is not true for the digit E, the smallest power of 2 starts with the digit E is indeed 221 = E2X20X8)
21XE = 59E18922E81631X39875663E89X853X91E595336X6114815X5X6929933X288E774E479575X628 may be the largest power of 2 not contain the digit 0, it has 65 digits.
The number 229 = 2368 (see power of 2#Powers of two whose exponents are powers of two) is very close to googol (10100), since it has EE digits. (thus, the Fermat number F9 (=229+1) is very close to googol) (it is a mathematical coincidence: 229 ≈ 10102)
1001 is the first four-digit palindromic number, and it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=13 + 103) = 509 + 6E4 (=93 + X3) (see taxicab number for other numbers), and it is also the smallest absolute Euler pseudoprime, note that there is no absolute Euler-Jacobi pseudoprime and no absolute strong pseudoprime. Since 1001 = 7×11×17, we can use the divisibility rule of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6k+1, 10k+1 and 16k+1 are all primes, then the product of them must be a Carmichael number (absolute Fermat pseudoprime), the smallest case is indeed 1001 (for k = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3X9).
All values of n > 45 (45 is the largest n such that this is not true, note that 45 is also the smallest prime that produces prime reciprocal magic square) for incrementally largest values of minimal x > 1 (or minimal y > 0) satisfying Pell's equation end with 1, and the dozens digit of all such values n > 2X1 are odd. (these values n are 2, 5, X, 11, 25, 3X, 45, 51, 91, 131, 1E1, 291, 2X1, 2E1, 391, 471, 711, 751, 971, X91, E31, ...)
The denominator of every nonzero Bernoulli number (except and ) ends with 6.
If n ends with 2 and n/2 is prime (or 1), then the denominator of the Bernoulli number is 6 (this is also true for some (but not all) n ends with X and n/2 is prime). (if the denominator of the Bernoulli number is 6, then n ends with 2 or X, but n/2 needs not to be prime or 1, the first counterexample is n = 82, the denominator of the Bernoulli number is 6, but 82/2 = 41 = 72 is neither prime nor 1)
is very close to 1.5, since a near-value for is 15/10 (=N4/P4, where Nn is nth NSW number, and Pn is nth Pell number, Nn/Pn is very close to when n is large). Besides, is very close to 2.2X, since a near-value for is 22X/100 (= L10/F10, where Ln is nth Lucas number, and Fn is nth Fibonacci number, Ln/Fn is very close to when n is large).
The recurring dozenal of the reciprocal of n terminates if and only if n is 3-smooth number (or harmonic number[1]) (i.e. n is regular to 10 if and only if n is 3-smooth number (or harmonic number)), since the 3-smooth numbers (or the harmonic numbers) are the numbers that evenly divide powers of 10. The 3-smooth numbers up to 1000 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, 23, 28, 30, 40, 46, 54, 60, 69, 80, 90, X8, 100, 116, 140, 160, 183, 194, 200, 230, 280, 300, 346, 368, 400, 460, 509, 540, 600, 690, 714, 800, 900, X16, X80, 1000. They are exactly the numbers k such that , where is the Euler's totient function. The sum of the reciprocals of the 3-smooth numbers is equal to 3, i.e. 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = 1 + 0.6 + 0.4 + 0.3 + 0.2 + 0.16 + 0.14 + 0.1 + ... = 3. Brief proof: 1/1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + ... = (Sum_{m>=0} 1/(2^m)) * (Sum_{n>=0} 1/(3^n)) = (1/(1-1/2)) * (1/(1-1/3)) = (2/(2-1)) * (3/(3-1)) = 3.
The 3-smooth numbers (or the numbers n such that the reciprocal of n terminates) ≤ 10 are 1, 2, 3, 4, 6, 8, 9, and 10, all of these numbers except 8 and 9 are divisors of 10 (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these n except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2), and the numbers 8 and 9 are in the Catalan's conjecture (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, (8, 9) is the largest pair of regular numbers which differ by 1, besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest Ziesel number and the smallest integer such that the factorization of over Q includes coefficients other than (i.e. the 89th cyclotomic polynomial, , is the first with coefficients other than ), besides, the repunit with length k (Rk) (where k = the concatenation of n and the unit (1), i.e. k = 10n+1) is prime for both n = 8 and n = 9, and not for any other n ≤ 1000, besides, the squares of 8 and 9 are the only two 2-digit automorphic numbers, besides, 8 and 9 are the only two natural numbers n such that centered n-gonal numbers (the kth centered n-gonal number is n×Tk+1, where Tk is the kth triangular number) cannot be primes (8 is because all centered 8-gonal numbers are square numbers (4-gonal numbers), 9 is because all centered 9-gonal numbers are triangular numbers (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, in fact, all polygonal numbers with rank > 2 are not primes, i.e. all primes p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number)), assuming the Bunyakovsky conjecture is true. (i.e. 8 and 9 are the only two natural number n such that is not irreducible) (Note that for n = 10, the centered 10-gonal numbers are exactly the star numbers)
The smallest n≥1 such that 4×60n−1 is prime (where 4 is the smallest composite number, and 60 is the smallest Achilles number, note that 4×60n is 3-smooth for all n≥1, thus 1/(4×60n) terminates in dozenal, and note that 4×60n is exactly 10000 (104) when n=2) is 460089, which contains the only four composite 3-smooth digits (4, 6, 8, 9) exactly once and from small to large in order (4 —> 6 —> 8 —> 9), and with two 0’s (zero digits) inside the middle, this prime (4×60460089−1) is 78E425-digit prime and proves the generalized Riesel problem base 60 (proves that 205 is the smallest generalized Riesel number in base 60, i.e. 205 is the smallest k such that gcd(k−1, 60−1) = 1 and k×60n−1 is composite for all n≥1, note that 205 = 4×60+4+1 = 4×(60+1)+1).
The 3-smooth numbers (or the numbers n such that the reciprocal of n terminates) ≤ 20 are 1, 2, 3, 4, 6, 8, 9, 10, 14, 16, and 20, all of these numbers are divisors of 100, note that the next two 3-smooth numbers (23 and 28) are not divisors of 100 (23 is because it has more prime factors 3 than 100, and 28 is because it has more prime factors 2 than 100) (thus, the numbers of digits of the reciprocal of all these n are all ≤2, while the numbers of digits of the reciprocal of 23 and 28 are 3).
Regular n-gon is constructible using neusis, or an angle trisector if and only if the reciprocal of is terminating number (where is Euler's totient function) (i.e. is 3-smooth, or is regular to 10), thus the n ≤ 1000 such that regular n-gon is constructible using neusis, or an angle trisector are 3, 4, 5, 6, 7, 8, 9, X, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 2X, 2E, 30, 31, 32, 33, 34, 36, 39, 40, 43, 44, 46, 48, 49, 50, 53, 54, 55, 58, 5X, 60, 61, 62, 64, 66, 68, 69, 70, 71, 76, 77, 7E, 80, 81, 86, 88, 89, 90, 91, 93, 94, 96, 99, 9E, X0, X6, X8, XX, E1, E3, E4, E8, 100, 102, 104, 108, 109, 110, 114, 116, 117, 120, 122, 123, 130, 132, 135, 139, 13X, 140, 141, 142, 143, 150, 154, 156, 160, 162, 163, 165, 166, 168, 170, 176, 17X, 180, 183, 187, 190, 193, 194, 195, 197, 198, 1X2, 1X6, 1X8, 1X9, 1E4, 1E9, 200, 203, 204, 208, 214, 216, 220, 223, 228, 22E, 230, 232, 233, 239, 240, 244, 246, 253, 259, 260, 264, 265, 26X, 276, 278, 280, 282, 284, 286, 293, 299, 2X0, 2X8, 2E0, 300, 301, 304, 306, 30X, 310, 314, 31E, 320, 323, 330, 338, 340, 341, 345, 346, 347, 349, 352, 360, 366, 367, 368, 369, 36X, 372, 374, 384, 390, 394, 395, 396, 3X3, 3X8, 3E3, 3E6, 400, 401, 403, 406, 408, 409, 414, 417, 428, 430, 440, 445, 446, 454, 45X, 460, 464, 466, 469, 473, 475, 476, 480, 487, 488, 490, 4X6, 4X7, 4E6, 500, 508, 509, 50X, 518, 519, 530, 534, 537, 539, 540, 541, 543, 544, 548, 549, 550, 566, 576, 57E, 580, 583, 594, 5X0, 5E3, 600, 602, 608, 609, 610, 618, 620, 628, 63X, 640, 646, 660, 669, 671, 674, 680, 682, 685, 689, 68X, 690, 692, 696, 699, 6X4, 6E3, 700, 710, 712, 714, 716, 718, 724, 728, 739, 748, 753, 760, 768, 76X, 770, 773, 781, 786, 794, 7X6, 7E0, 7E1, 800, 801, 802, 806, 810, 814, 816, 828, 832, 839, 853, 854, 860, 86E, 875, 880, 88X, 890, 891, 8X8, 8E1, 8E8, 8EE, 900, 901, 903, 908, 910, 916, 926, 92X, 930, 940, 947, 952, 954, 959, 960, 969, 977, 990, 992, 9X1, 9E0, X00, X03, X13, X14, X16, X17, X18, X19, X23, X34, X36, X60, X68, X72, X76, X79, X80, X82, X83, X86, X88, X8E, X94, X96, XX0, E10, E27, E30, E3X, E40, E43, E46, E55, E68, E69, E80, E99, EX6, 1000
If and only if n is a divisor of 20, then m2 = 1 mod n for every integer m coprime to n.
If and only if n is a divisor of 20, then the Dirichlet characters mod n are all real.
If and only if n is a divisor of 20, then n is divisible by all numbers less than or equal to the square root of n.
If and only if n is a divisor of 20, then k−1 is prime for all divisors k>2 of n.
If and only if n+1 is a divisor of 20, then is squarefree for all 0 ≤ k ≤ n, i.e. all numbers in the nth row of the Pascal's triangle are squarefree (the topmost row (i.e. the row which contains only one 1) of the Pascal's triangle is the 0th row, not the 1st row). (Note that all such n are primes or 1 or 0, and 20 is the largest number m such that if n+1 is a divisor of m, then n is prime or 1 or 0, besides, if and only if m is a divisor of 20, then m satisfies this condition)
If we only have the numbers 1 to 20 (including 1 and 20), then only the primes dividing 10 (i.e. primes ≤3) can be squared, since 5^2 = 21 > 20, and for the numbers such that the reciprocal of n terminates, they can only have at most 2 digits (which is the case of 8, 9, 14, 16 and 20), since the numbers with terminate reciprocal with >2 digits, they must be divisible by either 2^5 = 28 or 3^3 = 23, but both are >20. (these (prime power) numbers are >20: (prime > 3)^(>1), (odd prime)^(>2), 2^(>4))
20 is the largest integer that is divisible by all natural numbers no larger than its square root.
The exponents on the right hand side of
are exactly the numbers n such that 20n+1 is square. (note that 10 is one of such numbers)
20 is the Euler characteristic of a K3 surface.
20 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: πn+3(Sn) = Z/20Z for all n ≥ 5.
20 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 10, 20 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/20Z)× = {±1, ±5, ±7, ±E} is isomorphic to the additive group (Z/2Z)3. This fact plays a role in monstrous moonshine.
The kissing number is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) except 6 and 10
The densest packing is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) but not the composite divisors of 10 (i.e. 4, 6, 10)
If xy≤20, then at least one of x and y is a divisor of 10 (this is not true for xy=21, 5×5=21, but 5 is not a divisor of 10).
googol (mod n) = googolplex (mod n) for all 1 ≤ n ≤ 20 (but not for n = 21).
For all numbers n ≤ 100 (but not for n = 101, and not for n = smallest prime > 100 (i.e. 105)), there is k ≤ 6 such that nk−1 or nk+1 (or both) is prime. (note that for n = 101, k = 7, 8 and 9 also not satisfy this condition, the smallest k satisfying this condition for n = 101 is X)
20 is the GCD of all Fermat-Wilson quotients. (thus, Fermat-Wilson quotients are never primes)
20 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 20 such spheres form the cells of a 20-cell), for 3-dimensional space, the kissing number is 10, the centers of 10 such spheres form the faces of a 10-hedron.
20! is very close to Avogadro constant, which is defined by 1/10 (10%) of the mass of one carbon-10 atom.
20-karat gold is pure, 16-karat gold is 16 parts gold, 6 parts another metal (forming an alloy with 90% gold), 12-karat gold is 12 parts gold, X parts another metal (forming an alloy with 70% gold), and so forth.
100 is the smallest number with an equal number of deficient and abundant divisors.
"100+n is prime if and only if 100−n is prime" is true for all n≤40 end with 1 or E except n = 1E (which 100+1E = 11E is prime, but 100−1E = 100−20+1 = 102−2×10+1 = (10−1)2, which has algebraic factors and cannot be prime).
Both 10 and 100 are equal to the sum of (the largest Fermat prime below it) and (the largest Mersenne prime below it) (10 = 5+7, 100 = 15+X7).
10 is the smallest base such that “the largest known Mersenne prime p such that 2^p−1 is also (Mersenne) prime” (X7) does not contain the digit 1 (note that X7 is also the largest n such that there is no k≤n such that n×k±1 are twin primes, and the smallest k such that n×k±1 are twin primes for n=X7 is exactly 100, besides, X7 is also the largest known n such that the Mersenne number 2^n−1 and the Wagstaff number (2^n+1)/3 are both primes, The New Mersenne Conjecture is that “n is of the form 2^k±1 or 4^k±3 (or both)” + “2^n−1 is prime” + “(2^n+1)/3 is prime” is never 2, and X7 is conjectured to be the largest n such that this number is 3, the second-largest such number is 51, and 51 and 51×2 (=X2) are also the second-largest and the third-largest n such that there is no k≤n such that n×k±1 are twin primes, 51 is also the largest n such that there is no k≤2n (also k≤4n) such that n×k±1 are twin primes).
100 followed by 100 1’s is prime.
100 is the smallest number whose nth power can be written as the sum of (≥2 and <n) positive nth powers for some n (n=5, the formula is 100^5 = 23^5 + 70^5 + 92^5 + E1^5, only four 5th powers).
100 is the smallest number whose 5th power can be written as the sum of 4 positive 5th powers, besides, the smallest number whose 5th power can be written as the sum of 5 positive 5th powers is 60, which is exactly half of 100, besides, the smallest number whose 5th power can be written as the sum of 6 positive 5th powers is 10, which is exactly square root of 100
100 is the maximum number of steps of numbers < 4X7 (the smallest number that reach a number > 5414 (the record which gotten by the start value 23), namely 100E74) in Collatz sequence (for the numbers 461, 466, 467 and 477).
All composite numbers ≤200 has at least one prime factor ≤10+1 (this is not true for the next number (201)).
All composite numbers ≤200 has at least one palindromic prime factor (this is not true for the next number (201)).
All composite numbers ≤200 has at least one repdigit prime factor (this is not true for the next number (201)).
200 = (smallest n such that 195×2^n+1 is prime) (1E3) + (smallest n such that 2^n+195 is prime) (9). (195 is the second-largest known Fermat prime, and the only known Fermat prime which is irregular prime, and the only known Fermat prime p such that p+2 is not prime) (note that for the largest known Fermat prime 31E15, smallest n such that 31E15×2^n+1 is prime is 200−1 (=1EE), and 2+31E15 is prime (in fact, for all known Fermat primes p except 195, 2+p is also prime), thus for p=31E15, (smallest n such that 31E15×2^n+1 is prime) (1EE) + (smallest n such that 2^n+31E15 is prime) (1) is also 200)
The exponents on the right hand side of are exactly the numbers n such that kn+1 is square for k=20. (note that 10 is one of such numbers)
All negative-Pell solvable numbers (i.e. numbers n such that x2−ny2 = −1 is solvable) end with negative-Pell solvable digits (i.e. end with 1, 2, 5 or X).
By Benford's law, the probability for the leading digit d (d ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}) occurs (for some sequences, e.g. powers of 2 (1, 2, 4, 8, 14, 28, 54, X8, 194, 368, 714, 1228, 2454, ...) and Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 11, 19, 2X, 47, 75, 100, ...)) are:
d | probability | d | probability |
---|---|---|---|
1 | 34.2% | 7 | 7.9% |
2 | 1E.6% | 8 | 6.X% |
3 | 14.8% | 9 | 6.1% |
4 | 10.E% | X | 5.6% |
5 | X.7% | E | 5.1% |
6 | 8.E% |
(Note: the percentage in the list are also in dozenal, i.e. 20% means 0.2 or , 36% means 0.36 or , 58.7% means 0.587 or )
Star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by 1 (the triangular numbers are 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, ..., and the star numbers are 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, ...), thus, all star numbers end with 1. (The star numbers are exactly the centered 10-gonal numbers)
The Hilbert numbers are the numbers end with 1, 5 or 9. (i.e. = 1 mod 4)
The Lagado numbers are the numbers end with 1, 4, 7 or X. (i.e. = 1 mod 3)
The smallest two 4-digit palindromic numbers (1001 and 1111) are both Ziesel numbers, they are also the smallest two palindromic numbers which cannot be prime when read in any base, and the smallest 4-digit palindromic number (1001) is exactly the smallest absolute Euler pseudoprime and the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=13 + 103) = 509 + 6E4 (=93 + X3), besides, 1111 is also the only known n such that both n and n+1 are Quasi-Carmichael numbers (squarefree composites n with the property that for every prime factor p of n, p+b divides n+b positively with b being any integer besides 0), besides, the smallest number needing >2 steps in the "Reverse and Add!" problem is 6E, and the palindrome at which it stops is exactly 1111 (6E + E6 = 165, 165 + 561 = 706, 706 + 607 = 1111), besides, 1111 is the smallest composite repunit number, besides, the square of 1111 is 1234321, another palindrome.
The largest 4-digit number (EEEE) is a member of a betrothed number pair (its betrothed number is 5600 (also a 4-digit number, note that 5600 is E-smooth), and if we calculate EEEE/gcd(EEEE, 5600), we get the 4-digit repunit (1111)).
“the smallest (Fermat) pseudoprime to both base 2 and base 3 that is not Carmichael number” (1691) and “conjectured largest panconsummate number” (1961) are both strobogrammatic numbers (the same upside down), and a number written backwards is another number. (these two numbers are both composite, however, the strobogrammatic numbers 160091 and 190061 are primes, also a number written backwards is another number, thus they are also emirps) (of course, the Zeisel numbers 1001 and 1111 are also strobogrammatic numbers)
All prime numbers end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or E), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or E (1 and all prime digits that do not divide 10), since all prime numbers other than 2 and 3 are coprime to 10.
The density of primes end with 1 is relatively low, but the density of primes end with 5, 7 and E are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or E)
Except (3, 5), all twin primes end with (5, 7) or (E, 1), and the density of these two types of twin primes are nearly equal.
The sum of any pair of twin primes (other than (3, 5)) ends with 0.
If n ≥ 3 and n is not divisible by E, then there are infinitely many primes with digit sum n.
All palindromic primes except 11 has an odd number of digits, since all even-digit palindromic numbers are divisible by 11. The palindromic primes below 1000 are 2, 3, 5, 7, E, 11, 111, 131, 141, 171, 181, 1E1, 535, 545, 565, 575, 585, 5E5, 727, 737, 747, 767, 797, E1E, E2E, E6E
All lucky numbers end with digit 1, 3, 7 or 9.
Except for 3, all Fermat primes end with 5. (In fact, there are only 5 known Fermat primes (3, 5, 15, 195 and 31E15) and it is conjectured that there are no more Fermat primes, interestingly, all digits of all known Fermat primes are odd)
Except for 3, all Mersenne primes end with 7. (Besides, all Mersenne primes except 3 and 7 end with one of the only two 2-digit Mersenne primes (27 and X7))
Except for 2 and 3, all Sophie Germain primes end with 5 or E.
Except for 5 and 7, all safe primes end with E.
A prime p is Gaussian prime (prime in the ring , where ) if and only if p ends with 7 or E (or p=3). (i.e. p = 3 mod 4)
A prime p is Eisenstein prime (prime in the ring , where ) if and only if p ends with 5 or E (or p=2). (i.e. p = 2 mod 3)
A prime p can be written as x2 + y2 if and only if p ends with 1 or 5 (or p=2). (i.e. p = 1 or 2 mod 4)
A prime p can be written as x2 + 3y2 if and only if p ends with 1 or 7 (or p=3). (i.e. p = 0 or 1 mod 3)
A number n can be written as x2 + y2 if and only if all prime factors of the squarefree part of n end with 1 or 5 (or 2).
A number n can be written as x2 + 3y2 if and only if all prime factors of the squarefree part of n end with 1 or 7 (or 3).
All numbers ≤ 20 coprime to 10 are either primes or 1 (unit). (this is not true for 21, 21 is the smallest composite coprime to 10)
All full reptend primes end with 5 or 7. (in fact, for all primes p ≥ 5, (p-1)/(the period length of 1/p) is odd if and only if p is end with 5 or 7, since 10 is a quadratic nonresidue mod p (i.e. , where is the Legendre symbol) if and only if p is end with 5 or 7, by quadratic reciprocity, and if 10 is a quadratic residue mod a prime, then 10 cannot be a primitive root mod this prime) However, the converse is not true, 17 is not a full reptend prime, since the recurring digits of 1/17 is 0.076E45076E45..., which has only period 6. If and only if p is a full reptend prime, then the recurring digits of 1/p is cyclic number, e.g. the recurring digits of 1/5 is the cyclic number 2497 (the cyclic permutations of the digits are this number multiplied by 1 to 4), and the recurring digits of 1/7 is the cyclic number 186X35 (the cyclic permutations of the digits are this number multiplied by 1 to 6). The full reptend primes below 1000 are 5, 7, 15, 27, 35, 37, 45, 57, 85, 87, 95, X7, E5, E7, 105, 107, 117, 125, 145, 167, 195, 1X5, 1E5, 1E7, 205, 225, 255, 267, 277, 285, 295, 315, 325, 365, 377, 397, 3X5, 3E5, 3E7, 415, 427, 435, 437, 447, 455, 465, 497, 4X5, 517, 527, 535, 545, 557, 565, 575, 585, 5E5, 615, 655, 675, 687, 695, 6X7, 705, 735, 737, 745, 767, 775, 785, 797, 817, 825, 835, 855, 865, 8E5, 8E7, 907, 927, 955, 965, 995, 9X7, 9E5, X07, X17, X35, X37, X45, X77, X87, X95, XE7, E25, E37, E45, E95, E97, EX5, EE5, EE7. (Note that for the primes end with 5 or 7 below 30 (5, 7, 15, 17, 25 and 27, all numbers end with 5 or 7 below 30 are primes), 5, 7, 15 and 27 are full reptend primes, and since 5×25 = 101 = , the period of 25 is 4, which is the same as the period of 5, and we can use the test of the divisiblity of 5 to test that of 25 (form the alternating sum of blocks of two from right to left), and since 7×17 = E1 = , the period of 17 is 6, which is the same as the period of 7, and we can use the test of the divisiblity of 7 to test that of 17 (form the alternating sum of blocks of three from right to left), thus, 17 and 25 are not full reptend primes, and they are the only two non-full reptend primes end with 5 or 7 below 30)
By Midy theorem, if p is a prime with even period length (let its period length be n), then if we let , then ai + ai+n/2 = E for every 1 ≤ i ≤ n/2. e.g. 1/5 = 0.249724972497..., and 24 + 97 = EE, and 1/7 = 0.186X35186X35..., and 186 + X35 = EEE, all primes (other than 2 and 3) ≤ 37 except E, 1E and 31 have even period length, thus they can use Midy theorem to get an E-repdigit number, the length of this number is the period length of this prime. (see below for the recurring digits for 1/n for all n ≤ 30)
The unique primes below 1060 are E, 11, 111, E0E1, EE01, 11111, 24727225, E0E0E0E0E1, E00E00EE0EE1, 100EEEXEXEE000101, 1111111111111111111, EEEE0000EEEE0000EEEE0001, 100EEEXEE0000EEEXEE000101, 10EEEXXXE011110EXXXE00011, EEEEEEEE00000000EEEEEEEE00000001, EEE000000EEE000000EEEEEE000EEEEEE001, and the period length of their reciprocals are 1, 2, 3, X, 10, 5, 18, 1X, 19, 50, 17, 48, 70, 5X, 68, 53.
If p is a safe prime other than 5, 7 and E, then the period length of 1/p is (p-1)/2. (this is not true for all primes ends with E (other than E itself), the first counterexample is p = 2EE, where the period length of 1/p is only 37)
There is no full reptend prime ends with 1, since 10 is quadratic residue for all primes end with 1. (if so, then this prime p is a proper prime (i.e. for the reciprocal of such primes (1/p), each digit 0, 1, 2, ..., E appears in the repeating sequence the same number of times as does each other digit (namely, (p−1)/10 times)), see repeating decimal#Fractions with prime denominators) (In fact, not only for base 10 such primes do not exist, for all bases = 0 mod 4 (i.e. bases end with digit 0, 4 or 8), such primes do not exist)
5 and 7 are the only two safe primes which are also full reptend primes, since except 5 and 7, all safe primes end with E, and 10 is quadratic residue for all primes end with E. (if so, then this prime p produces a stream of p−1 pseudo-random digits, see repeating decimal#Fractions with prime denominators) (In fact, not only for base 10 there are only finitely many such primes, of course for square bases (bases of the form k2) only 2 may be full reptend prime (if the base is odd), and all odd primes are not full reptend primes, but since all safe primes are odd primes, for these bases such primes do not exist, besides, for the bases of the form 3k2, only 5 and 7 can be such primes, the proof for these bases is completely the same as that for base 10)
p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p | p | period length of 1/p |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 0 | 111 | 3 | 267 | 266 | 41E | 20E | 591 | 3X | 767 | 766 | 927 | 926 | E1E | 56E | 1107 | 1106 | 12E5 | 12E4 | 14E1 | 14E | 16X7 | 16X6 |
3 | 0 | 117 | 116 | 271 | 27 | 421 | 63 | 59E | 2XE | 76E | 395 | 955 | 954 | E21 | 570 | 1115 | 1114 | 1301 | 760 | 14E5 | 14E4 | 16E5 | 188 |
5 | 4 | 11E | 6E | 277 | 276 | 427 | 426 | 5E1 | 159 | 771 | 132 | 95E | 48E | E25 | E24 | 1125 | 1124 | 1317 | 506 | 14EE | 85E | 16E7 | 16E6 |
7 | 6 | 125 | 124 | 27E | 13E | 431 | 109 | 5E5 | 5E4 | 775 | 774 | 965 | 964 | E2E | 575 | 112E | 675 | 1337 | 512 | 150E | 865 | 1705 | 398 |
E | 1 | 12E | 75 | 285 | 284 | 435 | 434 | 5E7 | 66 | 77E | 39E | 971 | 172 | E31 | 116 | 1135 | 1134 | 133E | 77E | 1517 | 1XX | 1711 | 493 |
11 | 2 | 131 | 76 | 291 | 83 | 437 | 436 | 5EE | 2EE | 785 | 784 | 987 | 32X | E37 | E36 | 114E | 685 | 1345 | 1344 | 1521 | 436 | 1715 | 1714 |
15 | 14 | 13E | 7E | 295 | 294 | 447 | 446 | 611 | 163 | 791 | 3X6 | 995 | 994 | E45 | E44 | 1151 | 115 | 1351 | 166 | 1525 | 1524 | 1727 | 1726 |
17 | 6 | 141 | 20 | 2X1 | 150 | 455 | 454 | 615 | 614 | 797 | 796 | 9X7 | 9X6 | E61 | 16 | 1165 | 1164 | 1365 | 1364 | 1547 | 1E2 | 1735 | 1734 |
1E | E | 145 | 144 | 2XE | 155 | 457 | 15X | 617 | 206 | 7X1 | 138 | 9XE | 4E5 | E67 | 3X2 | 1167 | 42 | 1367 | 1366 | 1561 | 89 | 1745 | 1744 |
25 | 4 | 147 | 56 | 2E1 | 26 | 45E | 22E | 61E | 30E | 7EE | 3EE | 9E1 | 9E | E6E | 595 | 1185 | 1184 | 136E | 795 | 156E | 97 | 1747 | 1746 |
27 | 26 | 157 | 12 | 2EE | 37 | 465 | 464 | 637 | 212 | 801 | 140 | 9E5 | 9E4 | E71 | 596 | 118E | 6X5 | 1377 | 106 | 1577 | 1576 | 1751 | 32X |
31 | 9 | 167 | 166 | 301 | 90 | 46E | 7 | 63E | 31E | 80E | 405 | 9EE | 4EE | E91 | 2E3 | 1197 | 472 | 138E | 7X5 | 157E | 89E | 1755 | 1754 |
35 | 34 | 16E | 95 | 307 | 102 | 471 | 13 | 647 | 216 | 817 | 816 | X07 | X06 | E95 | E94 | 11X1 | 6E | 1391 | 3E3 | 1585 | 1584 | 1757 | 1756 |
37 | 36 | 171 | 96 | 30E | 165 | 481 | 24 | 655 | 654 | 825 | 824 | X0E | 505 | E97 | E96 | 11X5 | 11X4 | 1395 | 1394 | 1587 | 2X | 176E | 995 |
3E | 1E | 175 | 8 | 315 | 314 | 485 | 44 | 661 | 176 | 82E | 415 | X11 | 56 | EX5 | EX4 | 11X7 | 11X6 | 13X1 | 7E0 | 1591 | 2E6 | 1781 | 4E0 |
45 | 44 | 17E | 9E | 321 | 170 | 48E | 245 | 665 | 138 | 835 | 834 | X17 | X16 | EE5 | EE4 | 11XE | 6E5 | 13X7 | 536 | 15XE | 8E5 | 1785 | 1784 |
4E | 25 | 181 | X0 | 325 | 324 | 497 | 496 | 66E | 335 | 841 | 84 | X27 | 156 | EE7 | EE6 | 11E7 | 11E6 | 13E1 | 13E | 15EE | 8EE | 178E | 9X5 |
51 | 13 | 18E | X5 | 327 | 10X | 4X5 | 4X4 | 675 | 674 | 851 | 14X | X35 | X34 | 1005 | 1004 | 1201 | 700 | 13E5 | 13E4 | 1601 | 160 | 1797 | 1796 |
57 | 56 | 195 | 194 | 32E | 175 | 4E1 | 9X | 687 | 686 | 855 | 854 | X37 | X36 | 1011 | 73 | 120E | 705 | 1405 | 1404 | 1615 | 1614 | 17X1 | 9E0 |
5E | 2E | 19E | XE | 33E | 17E | 4EE | 25E | 68E | 345 | 85E | 42E | X3E | 51E | 1017 | 1016 | 1211 | 706 | 1407 | 326 | 1621 | 910 | 17X5 | 17X4 |
61 | 30 | 1X5 | 1X4 | 347 | 46 | 507 | 182 | 695 | 694 | 865 | 864 | X41 | 188 | 1021 | 610 | 121E | 70E | 1425 | 1424 | 1625 | 1624 | 17EE | 9EE |
67 | 22 | 1X7 | 46 | 34E | 2E | 511 | 266 | 69E | 34E | 867 | 2X2 | X45 | X44 | 1027 | 1026 | 1231 | 123 | 142E | 815 | 1635 | 274 | 1807 | 682 |
6E | 35 | 1E1 | E6 | 357 | 11X | 517 | 516 | 6X7 | 6X6 | 871 | 152 | X4E | 525 | 1041 | 620 | 123E | 71E | 1431 | 286 | 1647 | 276 | 1815 | 1814 |
75 | 8 | 1E5 | 1E4 | 35E | 18E | 51E | 45 | 6E1 | 6E | 881 | 440 | X5E | 52E | 1047 | 1046 | 1245 | 1244 | 1437 | 1436 | 1655 | 1654 | 181E | X0E |
81 | 14 | 1E7 | 1E6 | 365 | 364 | 527 | 526 | 701 | 360 | 88E | 445 | X6E | 535 | 104E | 625 | 1255 | 114 | 143E | 81E | 1657 | 61X | 1825 | 1824 |
85 | 84 | 205 | 204 | 375 | 34 | 531 | 276 | 705 | 704 | 8X5 | 98 | X77 | X76 | 1051 | 313 | 1257 | 49X | 1445 | 1444 | 165E | 92E | 1831 | 509 |
87 | 86 | 217 | 86 | 377 | 376 | 535 | 534 | 70E | 365 | 8X7 | 2E6 | X87 | X86 | 1061 | 16 | 125E | 72E | 1457 | 1456 | 1667 | 622 | 183E | X1E |
8E | 45 | 21E | 10E | 391 | 1X6 | 541 | 54 | 711 | 71 | 8XE | 455 | X91 | 283 | 106E | 635 | 1261 | 730 | 1461 | 38 | 1671 | 936 | 184E | X25 |
91 | 46 | 221 | 66 | 397 | 396 | 545 | 544 | 71E | 36E | 8E5 | 8E4 | X95 | X94 | 107E | 63E | 126E | 735 | 1465 | 1464 | 1677 | 20X | 1861 | 269 |
95 | 94 | 225 | 224 | 3X5 | 3X4 | 557 | 556 | 721 | 370 | 8E7 | 8E6 | X9E | 54E | 1087 | 1086 | 127E | 73E | 1467 | 562 | 167E | 93E | 1865 | 1864 |
X7 | X6 | 237 | 92 | 3XE | 1E5 | 565 | 564 | 727 | 24X | 901 | 230 | XX7 | 376 | 109E | 64E | 1281 | 740 | 1471 | 419 | 1681 | 140 | 186E | X35 |
XE | 55 | 241 | 120 | 3E5 | 3E4 | 575 | 574 | 735 | 734 | 905 | 198 | XXE | 555 | 10E1 | 329 | 1295 | 94 | 1475 | 1474 | 1685 | 1684 | 1875 | 1874 |
E5 | E4 | 24E | 125 | 3E7 | 3E6 | 577 | 116 | 737 | 736 | 907 | 906 | XE7 | XE6 | 10E7 | 10E6 | 1297 | 1296 | 147E | 83E | 168E | 945 | 1877 | 146 |
E7 | E6 | 251 | 73 | 401 | 100 | 585 | 584 | 745 | 744 | 90E | 465 | XEE | 55E | 10EE | 65E | 12X1 | 75 | 148E | 845 | 1697 | 1696 | 189E | X4E |
105 | 104 | 255 | 254 | 40E | 205 | 587 | 1XX | 747 | 9X | 91E | 46E | E11 | 1X2 | 1101 | 220 | 12X5 | 2E8 | 1495 | 1494 | 169E | 94E | 18X1 | 210 |
107 | 106 | 25E | 12E | 415 | 414 | 58E | 2X5 | 751 | 1X3 | 921 | 236 | E15 | 228 | 1105 | 1104 | 12X7 | 12X6 | 149E | 84E | 16X1 | 950 | 18XE | X55 |
period length | primes | period length | primes |
---|---|---|---|
1 | E | 11 | 1E0411, 69X3901 |
2 | 11 | 12 | 157, 7687 |
3 | 111 | 13 | 51, 471, 57E1 |
4 | 5, 25 | 14 | 15, 81, 106X95 |
5 | 11111 | 15 | X9X9XE, 126180EE0EE |
6 | 7, 17 | 16 | E61, 1061 |
7 | 46E, 2X3E | 17 | 1111111111111111111 |
8 | 75, 175 | 18 | 24727225 |
9 | 31, 3X891 | 19 | E00E00EE0EE1 |
X | E0E1 | 1X | E0E0E0E0E1 |
E | 1E, 754E2E41 | 1E | 3E, 78935EX441, 523074X3XXE |
10 | EE01 | 20 | 141, 8E5281 |
The period level of a prime p ≥ 5 is (p−1)/(period length of 1/p), e.g., has period level 3, thus the numbers with integer 1 ≤ a ≤ 16 from 3 different cycles: 076E45 (for a = 1, 7, 8, E, 10, 16), 131X8X (for a = 2, 3, 5, 12, 14, 15) and 263958 (for a = 4, 6, 9, X, 11, 13). Besides, has period level 1, thus this number is a cyclic number and 15 is a full-reptend prime, and all of the numbers with integer 1 ≤ a ≤ 14 from the cycle 08579214E36429X7.
There are only 9 repunit primes below R1000: R2, R3, R5, R17, R81, R91, R225, R255 and R4X5 (Rn is the repunit with length n). If p is a Sophie Germain prime other than 2, 3 and 5, then Rp is divisible by 2p+1, thus Rp is not prime. (The length for the repunit (probable) primes are 2, 3, 5, 17, 81, 91, 225, 255, 4X5, 5777, 879E, 198E1, 23175, 311407, ..., note that 879E is the smallest (and the only known) such number ends with E)
A prime p divides Rp−1 (where Rn is the repunit with length n) if and only if p2 (the square of p) is not Brazilian number (numbers which are nontrivial repdigit to some base, these numbers are also highly related to the generalized repunit numbers). (E is the only prime p not dividing Rp−1, E is also the only prime p such that p2 is Brazilian number (E2 = X1 = 111113))
The repunit number 111 is exactly the largest known number n such that (nn+1)/(n+1) (the generalized repunit number with length n in negative base −n) is (probable) prime.
By Fermat's little theorem, if p is a prime other than 2, 3 and E, then p divides the repunit with length p−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54, the counterexamples up to 1000 are 55, 77, E1, 101, 187, 275, 4X7, 777, 781, E55, they are exactly the Fermat pseudoprimes for base 10 (composite numbers c such that 10c-1 = 1 mod c) which are not divisible by E, they are called "deceptive primes", if n is deceptive prime, then Rn is also deceptive prime, thus there are infinitely may deceptive primes) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's.
n | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +X | +E | +10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
0+ | 1 | 10 | 10 | 10 | 101 | 10 | 1001 | 100 | 100 | 1010 | 11111111111 | 10 |
10+ | 11 | 10010 | 1010 | 100 | 10111 | 100 | 1001 | 1010 | 10010 | 111111111110 | 11101 | 100 |
20+ | 110111 | 110 | 1000 | 10010 | 101 | 1010 | 101011 | 1000 | 111111111110 | 101110 | 101101 | 100 |
30+ | 1001001 | 10010 | 110 | 10100 | 111001 | 10010 | 101001 | 111111111110 | 10100 | 111010 | 10001111 | 100 |
40+ | 10111101 | 1101110 | 101110 | 110 | 1100101 | 1000 | 1011101111111 | 100100 | 10010 | 1010 | 101011 | 1010 |
50+ | 10010101 | 1010110 | 100100 | 1000 | 1111 | 111111111110 | 1100101 | 101110 | 111010 | 1011010 | 1100111 | 100 |
60+ | 10101101 | 10010010 | 1101110 | 10010 | 1011101111111 | 110 | 10101011 | 10100 | 10000 | 1110010 | 100111001 | 10010 |
70+ | 1101001 | 1010010 | 1010 | 1111111111100 | 10001 | 10100 | 1001 | 111010 | 1010110 | 100011110 | 101101 | 1000 |
80+ | 111011 | 101111010 | 1111111111100 | 1101110 | 110001 | 101110 | 10100111 | 1100 | 1011010 | 11001010 | 100111 | 1000 |
90+ | 1010111111 | 10111011111110 | 10010010 | 100100 | 10101001 | 10010 | 110101001 | 1010 | 1100 | 1010110 | 101100011 | 10100 |
X0+ | 111111111101 | 100101010 | 1110010 | 1010110 | 11100001 | 100100 | 1100001 | 10000 | 1010010 | 11110 | 1111011111 | 111111111110 |
E0+ | 1001 | 11001010 | 101000 | 1011100 | 101011 | 111010 | 11010111 | 1011010 | 100011110 | 11001110 | 1111111111111111111111 | 100 |
n | 1 | 5 | 7 | E | 11 | 15 | 17 | 1E | 21 | 25 | 27 | 2E |
smallest k such that k×n is a repunit | 1 | 275 | 1X537 | 123456789E | 1 | 92X79E43715865 | 8327 | 69E63848E | 634X159788253X72E1 | 55 | 509867481E793XX5X1243628E317 | 45X3976X7E |
the length of the repunit k×n | 1 | 4 | 6 | E | 2 | 14 | 6 | E | 18 | 4 | 26 | 10 |
(this k is usually not prime, in fact, this k is not prime for all numbers n < 100 which are coprime to 10 except n = 55, and for n < 1000 which is coprime to 10, this k is prime only for n = 55, 101, 19E, 275 and 46E, and only 19E and 46E are itself prime, other 3 numbers are 5×11, 5×25 and 11×25, and this k for these n are successively 25, 11 and 5, which makes k×n = R4 = 1111 = 5×11×25, besides, this k for n = 46E is 2X3E, which makes k×n = R7 = 1111111, a repunit semiprime, and this k for n = 19E is a X8-digit prime number, with k×n = RXE, another repunit semiprime)
For every prime p except E, the repunit with length p is congruent to 1 mod p. (The converse is also not true, the counterexamples up to 1000 are 4, 6, 10, 33, 55, 77, E1, 101, 187, 1E0, 275, 444, 4X7, 777, 781, E55, they are called "repunit pseudoprimes" (or weak deceptive primes), all deceptive primes are also repunit pseudoprimes, if n is repunit pseudoprime, then Rn is also repunit pseudoprime, thus there are infinitely may repunit pseudoprimes. No repunit pseudoprimes are divisible by 8, 9 or E. (in fact, the repunit pseudoprimes are exactly the weak pseudoprimes for base 10 (composite numbers c such that 10c = 10 mod c) which are not divisible by E) Besides, the deceptive primes are exactly the repunit pseudoprimes which are coprime to 10)
Smallest multiple of n with digit sum 2 are: (0 if not exist)
- 2, 2, 20, 20, 101, 20, 1001, 20, 200, 1010, 0, 20, 11, 10010, 1010, 200, 100000001, 200, 1001, 1010, 10010, 0, 0, 20, 10000000001, 110, 2000, 10010, 101, 1010, 1000000000000001, 200, 0, 1000000010, 1000000000001, 200, ..., if and only if n is divisible by some prime p with 1/p odd period length, then such number does not exist.
Smallest multiple of n with digit sum 3 are: (0 if not exist)
- 3, 12, 3, 30, 21, 30, 12, 120, 30, 210, 0, 30, 0, 12, 210, 300, 201, 30, 10101, 210, 120, 0, 1010001, 120, 21, 0, 300, 120, 0, 210, 1010001, 1200, 0, 2010, 200001, 30, ..., such number does not exist for n divisible by E, 11 or 25.
Smallest multiple of n with digit sum 4 are: (0 if not exist)
- 4, 4, 13, 4, 13, 40, 103, 40, 130, 130, 0, 40, 22, 1030, 13, 40, 3001, 130, 2002, 130, 103, 0, 11101, 40, 10012, 22, 1300, 1030, 202, 130, 10003, 400, 0, 30010, 101101, 130, ..., such number is conjectured to exist for all n not divisible by E (of course, if n is divisible by E, then such number does not exist).
Smallest multiple of n with digit sum n are:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 1E0, 20E, 22X, 249, 268, 287, 2X6, 45X, 488, 4E6, 1EX, 8E4, 3EX0, 3EE, 23EX, 1899, XX8, 2E79, 4E96, 1EX9, 4XX8, 2EE9, 3XEX, 799X, 5EE90, ..., such number is conjectured to exist for all n.
45 is the smallest prime that produces prime reciprocal magic square, i.e. write the recurring digits of 1/45 (=0.2872E3X23205525X784640XX4E9349081989E6696143757E117, which has period 44) to 44/45, we get a 44×44 prime reciprocal magic square (its magic number is 1EX), it is conjectured that there are infinitely many such primes, but 45 is the only such prime below 1000, all such primes are full reptend primes, i.e. the reciprocal of them are cyclic numbers, and 10 is a primitive root modulo these primes.
All numbers of the form 34{1} are composite (proof: 34{1n} = 34×10n+(10n−1)/E = (309×10n−1)/E and it can be factored to ((19×10n/2−1)/E) × (19×10n/2+1) for even n and divisible by 11 for odd n). Besides, 34 was proven to be the smallest n such that all numbers of the form n{1} are composite. However, the smallest prime of the form 23{1} is 23{1E78}, it has E7X digits. The only other two n≤100 such that all numbers of the form n{1} are composite are 89 and 99 (the reason of 89 is the same as 34, and the reason of 99 is 99{1n} is divisible by 5, 11 or 25).
The only known of the form 1{0}1 is 11 (see generalized Fermat prime), these are the primes obtained as the concatenation of a power of 10 followed by a 1. If n = 1 mod 11, then all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are divisible by 11 and thus composite. Except 10, the smallest n not = 1 mod 11 such that all numbers obtained as the concatenation of a power of n (>1) followed by a 1 are composite was proven by EX, since all numbers obtained as the concatenation of a power of EX (>1) followed by a 1 are divisible by either E or 11 and thus composite. However, the smallest prime obtained as the concatenation of a power of 58 (>1) followed by a 1 is 10×582781E5+1, it has 459655 digits.
All numbers of the form 1{5}1 are composite (proof: 1{5n}1 = (14×10n+1−41)/E and it can be factored to (4×10(n+1)/2−7) × ((4×10(n+1)/2−7)/E) for odd n and divisible by 11 for even n).
The emirps below 1000 are 15, 51, 57, 5E, 75, E5, 107, 117, 11E, 12E, 13E, 145, 157, 16E, 17E, 195, 19E, 1X7, 1E5, 507, 51E, 541, 577, 587, 591, 59E, 5E1, 5EE, 701, 705, 711, 751, 76E, 775, 785, 7X1, 7EE, E11, E15, E21, E31, E61, E67, E71, E91, E95, EE5, EE7.
The non-repdigit permutable primes below 1010100 are 15, 57, 5E, 117, 11E, 5EEE (the smallest representative prime of the permutation set).
The non-repdigit circular primes below 1010100 are 15, 57, 5E, 117, 11E, 175, 1E7, 157E, 555E, 115E77 (the smallest representative prime of the cycle).
The first few Smarandache primes are the concatenation of the first 5, 15, 4E, 151, ... positive integers.
The only known Smarandache–Wellin primes are 2 and 2357E11.
There are 15 minimal primes, and they are 2, 3, 5, 7, E, 11, 61, 81, 91, 401, X41, 4441, X0X1, XXXX1, 44XXX1, XXX0001, XX000001.
The smallest weakly prime is 6E8XE77.
There are 824E1 left-truncatable primes, the largest of them is 28-digit 471X34X164259EX16E324XE8X32E7817.
There are 12E right-truncatable primes, the largest of them is X-digit 375EE5E515.
There are 2E primes which are both left-truncatable and right-truncatable. They have been called two-sided primes, and they are 2, 3, 5, 7, E, 25, 27, 35, 37, 3E, 57, 5E, 75, E5, E7, 25E, 315, 357, 35E, 375, 3E5, 3E7, 517, 51E, 575, 5E5, 5E7, 2557, 35E7, 5117, 511E, 51E7, 5E1E, 375E5, 511E7.
The only two base 10 Wieferich primes up to 1010 are 1685 and 5E685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685. (there is also a note for the only two known base 2 Wieferich primes (771 and 2047) minus 1 written in base 2, 8 (= 23) and 14 (= 24), 770 = 010001000100(2) = 444(14) is a repdigit in base 14, and 2046 = 110110110110(2) = 6666(8) is also a repdigit in base 8, see Wieferich prime#Binary periodicity of p − 1)
For the numbers between 5X0 (the smallest number divisible by all of the numbers 1 to 8) and 630 (the square of 26 = 5#) and end with 1, 5, 7 or E (the digits coprime to 10), all numbers whose dozen digit is odd are primes, and all numbers whose dozen digit is even are composites.
For all odd composites c up to 1000, there exists integer a such that GCD(a, c) = 1 and a(c−1)/2 is congruent to neither 1 nor −1 mod c (i.e. c is not an Euler pseudoprime base a), however, this is not true for c = 1001, 1001 is Euler pseudoprime to all bases coprime to itself, i.e. 1001 is an absolute Euler pseudoprime. (note that there are no “absolute Euler-Jacobi pseudoprime”, every composite number is Euler-Jacobi pseudoprime to at most 30% (i.e. 1/4) bases coprime to it)
There are 1, 2, 3, 5 and 6-digit (but not 4-digit) narcissistic numbers, there are totally 73 narcissistic numbers, the first few of which are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 25, X5, 577, 668, X83, 14765, 938X4, 369862, X2394X, ..., the largest of which is 43-digit 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E. (see OEIS: A161949)
The only two factorions are 1 and 2.
The only seven happy numbers below 1000 are 1, 10, 100, 222, 488, 848 and 884, almost all natural numbers are unhappy. All unhappy numbers get to one of these four cycles: {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {68, 84}, or one of the only two fixed points other than 1: 25 and X5.
If we use the sum of the cubes (instead of squares) of the digits, then every natural numbers get to either 1 or the cycle {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}. (for the example of the famous Hardy–Ramanujan number 1001 = 93 + X3, we know that this sequence with initial term 9X is 9X, 1001, 2, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200, 8, ...)
n | fixed points and cycled for the sequence for sum of n-th powers of the digits | length of these cycles |
---|---|---|
1 | {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {X}, {E} | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 |
2 | {1}, {5, 21}, {8, 54, 35, 2X, 88, X8, 118, 56, 51, 22}, {18, 55, 42}, {25}, {68, 84}, {X5} | 1, 2, X, 3, 1, 2, 1 |
3 | {1}, {8, 368, 52E, X20, 700, 247, 2X7, 947, 7X8, 10X7, 940, 561, 246, 200}, {577}, {668}, {6E5, E74, 100X}, {X83}, {11XX} | 1, 12, 1, 1, 3, 1, 1 |
4 | {1}, {X6X, 103X8, 8256, 35X9, 9EXE, 22643, E69, 1102X, 596X, X842, 8394, 6442, 1080, 2455}, {206X, 6668, 4754}, {3X2E, 12396, 472E, X02X, E700, 9X42, 98X9, 13902} | 1, 12, 3, 8 |
The harshad numbers up to 200 are 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 1X, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, X0, X1, E0, 100, 10X, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1X0, 1E0, 1EX, 200, although the sequence of factorials begins with harshad numbers, not all factorials are harshad numbers, after 7! (=2E00, with digit sum 11 but 11 does not divide 7!), 8X4! is the next that is not (8X4! has digit sum 8275 = E×8E7, thus not divide 8X4!). There are no 21 consecutive integers that are all harshad numbers, but there are infinitely many 20-tuples of consecutive integers that are all harshad numbers.
The Kaprekar numbers up to 10000 are 1, E, 56, 66, EE, 444, 778, EEE, 12XX, 1640, 2046, 2929, 3333, 4973, 5E60, 6060, 7249, 8889, 9293, 9E76, X580, X912, EEEE.
The Kaprekar's routine of any four-digit number which is not repdigit converges to either the cycle {3EE8, 8284, 6376} or the cycle {4198, 8374, 5287, 6196, 7EE4, 7375}, and the Kaprekar map of any three-digit number which is not repdigit converges to the fixed point 5E6, and the Kaprekar map of any two-digit number which is not repdigit converges to the cycle {0E, X1, 83, 47, 29, 65}.
n | Cycles for Kaprekar's routine for n-digit numbers | Length of these cycles | Number of these cycles |
---|---|---|---|
1 | {0} | 1 | 1 |
2 | {00}, {0E, X1, 83, 47, 29, 65} | 1, 6 | 2 |
3 | {000}, {5E6} | 1, 1 | 2 |
4 | {0000}, {3EE8, 8284, 6376}, {4198, 8374, 5287, 6196, 7EE4, 7375} | 1, 3, 6 | 3 |
5 | {00000}, {64E66, 6EEE5}, {83E74} | 1, 2, 1 | 3 |
6 | {000000}, {420X98, X73742, 842874, 642876, 62EE86, 951963, 860X54, X40X72, X82832, 864654}, {65EE56} | 1, X, 1 | 3 |
7 | {0000000}, {841E974, X53E762, 971E943, X64E652, 960EX53, E73E741, X82E832, 984E633, 863E754}, {962E853} | 1, 9, 1 | 3 |
8 | {00000000}, {4210XX98, X9737422, 87428744, 64328876, 652EE866, 961EE953, X8428732, 86528654, 6410XX76, X92EE822, 9980X323, X7646542, 8320X984, X7537642, 8430X874, X5428762, 8630X854, X540X762, X830X832, X8546632, 8520X964, X740X742, X8328832, 86546654}, {873EE744}, {X850X632} | 1, 20, 1, 1 | 4 |
The self numbers up to 600 are 1, 3, 5, 7, 9, E, 20, 31, 42, 53, 64, 75, 86, 97, X8, E9, 10X, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1X9, 1EX, 20E, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2XX, 2EE, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39X, 3XE, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48X, 49E, 4E0, 501, 512, 514, 525, 536, 547, 558, 569, 57X, 58E, 5X0, 5E1.
The Friedman numbers up to 1000 are 121=112, 127=7×21, 135=5×31, 144=4×41, 163=3×61, 368=86−3, 376=6×73, 441=(4+1)4, 445=54+4.
The Keith numbers up to 1000 are 11, 15, 1E, 22, 2X, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, XX, EE, 125, 215, 24X, 405, 42X, 654, 80X, 8X3, X59.
There are totally 71822 polydivisible numbers, the largest of which is 24-digit 606890346850EX6800E036206464. However, there are no E-digit polydivisible numbers contain the digits 1 to E exactly once each. (hence there are also no 10-digit polydivisible numbers using all the digits 0 to E exactly once, since if a number with digits abcdefghijkl is a 10-digit polydivisible number using all the digits 0 to E exactly once, then {a, b, c, d, e, f, g, h, i, j, k, l} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and then abcdefghijkl is divisible by 10, thus we have l = 0 (by divisibility rule of 10), and {a, b, c, d, e, f, g, h, i, j, k} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, thus a number with digits abcdefghijk is an E-digit polydivisible numbers using all the digits 1 to E exactly once). (proof: if a number with digits abcdefghijk is an E-digit polydivisible numbers using all the digits 1 to E exactly once, then {a, b, c, d, e, f, g, h, i, j, k} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and we have:
f = 6 (since abcdef is divisible by 6) (by divisibility rule of 6)
{d, h} = {4, 8} (since abcd is divisible by 4 and abcdefgh is divisible by 8 (thus by 4)) (by divisibility rule of 4)
{c, i} = {3, 9} (since abc is divisible by 3 and abcdefghi is divisible by 9 (thus by 3)) (by divisibility rule of 3)
{b, j} = {2, X} (since ab is divisible by 2 and abcdefghij is divisible by X (thus by 2)) (by divisibility rule of 2)
thus, we have {a, e, g, k} = {1, 5, 7, E}
Since abcdefgh is divisible by 8, thus gh is divisible by 8 (by divisibility rule of 8), and since {a, e, g, k} = {1, 5, 7, E}, thus g is odd, and h must be 4 (if h = 8 and g is odd, then gh is not divisible by 8), and since abcdefghi is divisible by 9, thus hi is divisible by 9 (by divisibility rule of 9), however, h = 4 and i is either 3 or 9, but neither 43 nor 49 is divisible by 9.
If we do not require the number formed by its first 8 digits divisible by 8, then there are 2 solutions: 1X98265E347 and 7298X65E341 (neither satisfies that the number formed by its first 8 digits is divisible by 4).
If we do not require the number formed by its first 9 digits divisible by 9, then there are 4 solutions: 1X38E694725, 7X981634E25, 7X98E654321, and EX987634125 (only 7X98E654321 satisfies that the number formed by its first 9 digits is divisible by 3).
The candidate Lychrel numbers up to 1000 are 179, 1E9, 278, 2E8, 377, 3E7, 476, 4E6, 575, 5E5, 674, 6E4, 773, 7E3, 872, 8E2, 971, 9E1, X2E, X3E, X5E, X70, XXE, XE0, E2X, E3X, E5X, EXX. The only suspected Lychrel seed numbers up to 1000 are 179, 1E9, X3E and X5E. However, it is unknown whether any Lychrel number exists. (Lychrel numbers only known to exist in these bases: E, 15, 18, 22 and all powers of 2)
Most numbers that end with 2 are nontotient (in fact, all nontotients < 58 except 2X end with 2), except 2 itself, the first counterexample is 92, which equals φ(X1) = φ(E2) and φ(182) = φ(2×E2), next counterexample is 362, which equals φ(381) = φ(1E2) and φ(742) = φ(2×1E2), there are only 9 such numbers ≤ 10000 (the number 2 itself is not counted), all such numbers (except the number 2 itself) are of the form φ(p2) = p(p−1), where p is a prime ends with E.
There is a known generalized Cullen prime for all bases b ≤ 10 (but not for next base b = 11). (no matter whether you require n ≥ b−1 or not)
There is a known generalized Woodall prime for all bases b ≤ 100 (but not for next base b = 101). (no matter whether you require n ≥ b−1 or not)
There is a known generalized Cullen prime for all even bases b ≤ 100 (but not for next even base b = 102) if you don’t require n ≥ b−1
There is a known generalized Carol prime for all even bases b ≤ (100×2 + 100÷2) (=260) (but not for next even base b = 262).
There is a known generalized Kynea prime for all even bases b ≤ (100×2) (=200) (but not for next even base b = 202).
The generalized minimal primes problem has at most one unsolved family for all bases b ≤ 20 (but not for b = 21). (there is one unsolved family for b = 15, 17 and 19, and there are no unsolved families for all other b ≤ 20, but for b = 21, there are 10 unsolved families)
There are no n≤100 which is nontotient, noncototient, and untouchable. (the smallest such n is indeed the smallest even number > 100, i.e. 102)
By sieve of Eratosthenes, we can cross out every composites ≤ 20 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3). (however, we cannot cross out the composite 21 by sieve the primes dividing 10 (i.e. the primes ≤3) (i.e. the primes 2 and 3))
By sieve of Eratosthenes, we can cross out every composites ≤ 200 by sieve to the prime 10+1 (=11). (however, we cannot cross out the composite 201 by sieve to the prime 10+1 (=11))
For all odd composites c ≤ 1000, there exists integer b coprime to c such that b(c−1)/2 ≠ ±1 (mod c) (i.e. c is not Euler pseudoprime base b). (this is not true for the composite c = 1001, 1001 is the smallest absolute Euler pseudoprime)
The Wagstaff numbers is prime for all odd primes p ≤ 20 (but not for p = next odd prime (25)).
There is a known odd generalized Wieferich prime for all prime bases p ≤ 20 (but not for p = next prime (25)).
{3}1 (i.e. 333333...3333331) cannot be prime if the number of 3’s is even, because of algebra factors (difference of two squares), besides, {3}11 (i.e. 333333...33333311) cannot be prime except the number 11 itself, because this number is divisible by 11 if the number of 3’s is even, and has algebra factors (difference of two squares) if the number of 3’s is odd.
The smallest Perrin pseudoprime is a near-repunit 111101 (note that this number is square: 3752), this number only contains five 1's and one 0 (no any digit >1), and this number plus 10 is the repunit with length 6, i.e. 111111, and this number is exactly the final 6 (half of 10) digits of 123456789XE×E (a 10-digit number).
The repunit prime 111 is not only the smallest irregular prime with irregular index >1, but also exactly the largest known negative base −n such that the repunit with length n is prime (i.e. (111^111+1)/(111+1) is prime, and 111 is the largest known such number), such number can be prime only for prime n ends with 1 or 5, except n = 3 (for positive base, the largest known such n is 444E, note that 444E is a near-repdigit prime, and with a repdigit (444) followed by the digit E, such number can be prime only for prime n ends with 7 or E, except n = 2 and n = 3)
23 and 233 (but not 2333, 23333, 233333, ..., and conjectured no other such numbers) (the ratio for these two numbers is very close to 10) are both starting value set new records for number of steps to reach 1 for the Collatz (3x+1) sequences, and the only two numbers < 4X7 (which is also a starting value set new records for maximum value for the Collatz (3x+1) sequences, like the number 23) whose records of number of steps to reach 1 are ≥ (the previous record + 10).
The ratio for the two smallest Weird numbers (5X and 598) is very close to 10
The smallest amicable number pair is (164,1E8), and the next number of them in the home prime problem are the same: 225E. (both of these two numbers get to the 9E-digit prime 51712E1X75561E754940E168440995028213825E76175409310343166811361353638X487E880858E7E8E024666872X98X3096E504E110578059E65 after 58 steps)
The “beast number” (666), when add a digit “1” before it and add another digit “1” after it, it become a palindromic square 16661, it is the smallest palindromic square whose square root is not palindrome (12E), it is also the smallest palindromic square depending on base.
If we let the musical notes in an octave be numbers in the cyclic group Z10: C=0, C#=1, D=2, Eb=3, E=4, F=5, F#=6, G=7, Ab=8, A=9, Bb=X, B=E (see pitch class and music scale) (thus, if we let the middle C be 0, then the notes in a piano are -33 to 40), then x and x+3 are minor third, x and x+4 are major third, x and x+7 are perfect fifth (thus, we can use 7x for x = 0 to E to get the five degree cycle), etc. (since an octave is 10 semitones, a minor third is 3 semitones, a major third is 4 semitones, and a perfect fifth is 7 semitones, etc.) (if we let an octave be 1, then a semitone will be 0.1, and we can write all 10 notes on a cycle, the difference of two connected notes is 26 degrees or radians) Besides, the x major chord (x) is {x, x+4, x+7} in Z10, and the x minor chord (xm) is {x, x+3, x+7} in Z10, and the x major 7th chord (xM7) is {x, x+4, x+7, x+E}, and the x minor 7th chord (xm7) is {x, x+3, x+7, x+X}, and the x dominant 7th chord (x7) is {x, x+4, x+7, x+X}, and the x diminished 7th triad (xdim7) is {x, x+3, x+6, x+9}, since the frequency of x and x+6 is not simple integer fraction, they are not harmonic, and this diminished 7th triad is corresponding the beast number 666 (three 6's) (also, x and x+6 are tritone, which is not harmonic). Besides, x major scale uses the notes {x, x+2, x+4, x+5, x+7, x+9, x+E}, and x minor scale uses the notes {x, x+2, x+3, x+5, x+7, x+8, x+X}. Besides, the frequency of x+10 is twice as that of x, the frequency of x+7 is 1.6 (=3/2) times as that of x, and the frequency of x+5 is 1.4 (=4/3) times as that of x, they are all simple integer fractions (ratios of small integers), and they all have at most one digit after the duodecimal point, and we can found that 1.610 = X9.8E5809 is very close to 27 = X8, since 217 = 2134X8 is very close to 310 = 217669, the simple frequency fractions found for the scales are only 0.6, 0.8, 0.9, 1.4, 1.6 and 2, however, since the frequency of x+10 is twice as that of x, thus the frequency of x+1 (i.e. a semitone higher than x) is (=20.1) times as that of x. Let f(x) be the frequency of x, then we have f(2)/f(0) = 9/8 (=1.16), f(4)/f(2) = X/9 (=1.14), and f(5)/f(4) = 14/13 (this number is very close to ), and thus we have that f(5)/f(0) = (9/8) × (X/9) × (14/13) = 4/3. Also, we can found that 20.5 is very close to 1.4, and 20.7 is very close to 1.6.
All orders of non-cyclic simple group end with 0 (thus, all orders of unsolvable group end with 0), however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group (thus, all groups with order < 50 are solvable), (50 is the order of the alternating group A5, which is a non-cyclic simple group, and thus an unsolvable group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not completely true (although this is true for all orders ≤ 14000), the smallest counterexample is 14X28, however, all such orders are divisible by 4 and either 3 or 5 (i.e. divisible by either 10 or 18), and all such orders have at least 3 distinct prime factors, by these conditions, the smallest possible such order is indeed 50 = 22 × 3 × 5, next possible such order is 70 = 22 × 3 × 7, however, by Sylow theorems, the number of Sylow 7-subgroups of all groups with order 70 (i.e. the number of subgroups with order 7 of all groups with order 70) is congruent to 1 mod 7 and divides 70, hence must be 1, thus the subgroup with order 7 is a normal subgroup of the group with order 70, thus all groups with order 70 have a nontrivial normal subgroup and cannot be simple groups)
The probability for rolling a 6 on a dice is 0.2 or 20%, and the probability for rolling at least one 6 on a dice in 3 rolls is 0.508 (less than one half or 60%), and the probability for rolling at least one 6 on a dice in 4 rolls is 0.6268 (more than one half or 60%), and the probability for rolling a "double 6" on two dices is 0.04 or 4%, and the probability for rolling at least one "double 6" on two dices in 20 rolls is 0.5X9190... (less than one half or 60%), and the probability for rolling at least one "double 6" on two dices in 21 rolls is 0.609685... (more than one half or 60%).
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E
appear in the repeating digits of 1/5 (exactly the even digits ≤ 5 (except 0 (the smallest digit)) and the odd digits ≥ 6 (except E (the largest digit)))
appear in the repeating digits of 1/7 (exactly the odd digits ≤ 5 and the even digits ≥ 6)
appear in the repeating digits of 1/11 (exactly the smallest digit (0) and the largest digit (E))
(note that the number of digits ≤5 is equal to the number of digits ≥6 (both are 6, which is equal to half of the base we used in this wiki (10)), and all digits are either ≤5 or ≥6, but not both)
(note that all of 5, 7, and 11 are primes, and they are the only three primes ≤ 10+1 (=11) and divides neither 10 nor 10−1 (=E))
253 = 33 × 11 = 33 × (10+1) 254 = 25 × E = 25 × (10−1)
1253 = 33 × 7 × E = 33 × (6+1) × (10−1) 1254 = 25 × 5 × 11 = 25 × (6−1) × (10+1)
253−1 = 0.004E0 254−1 = 0.004X
(Note that 6 is half of 10, and 2 and 3 are the only prime factors of 10 (also the only prime factors of 6), and 25 and 33 are the smallest power of 2 and 3 which does not divide 100, besides, 2, 3, 6−1, 6+1, 10−1, 10+1 are exactly the first primes)
All k ≤ 394 has at least one prime with 2 ≤ n ≤ 600, but....
395 = 5 × 91 (5 times a prime number), isn't prime until n = 1696, however, the smallest n ≥ 2 such that is prime is only 10 (10 is exactly the base of this wiki).
3X1 = 7 × 67 (7 times a prime number), isn't prime until n = 989, however, the smallest n ≥ 2 such that is prime is only 2 (2 is exactly the smallest possible base).
(Note that 395 is the only divisor d of 76X such that is not prime, 76X has 8 divisors, and for all divisors d except 395, is prime, thus the number 10^76X−1 is product of 7 unique primes (primes p which there is no other prime q such that the period length of the dozenal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q) and the composite number , and 76X is conjectured to be the largest number such that for all divisors d except one number, is prime (note that 10 is exactly the base of this wiki, and if we change the number 10 to other small numbers (≤20), then there is no number as large as 76X such that for all divisors d except one number, is prime), besides, 76X+1 (=76E) is prime, by Fermat's little theorem (every prime p not dividing b must divide for some d dividing p−1), since 76E > 10, 76E does not divide 10, thus 76E must divide for some d dividing 76X, however, since for all d dividing 76X except 395 is prime (and not equal to 76E), 76E can only divide , and the period length for 1/76E must be 395)
Besides, the primes 5 and 7 are