Repunit

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In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — the simplest form of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.[1]

A repunit prime is a repunit that is also a prime number. In binary, these are the widely known Mersenne primes.

Definition[edit]

The base-b repunits are defined as

R_n^{(b)}={b^n-1\over{b-1}}\qquad\mbox{for }b\ge2, n\ge1.

Thus, the number Rn(b) consists of n copies of the digit 1 in base b representation. The first two repunits base b for n=1 and n=2 are

R_1^{(b)}={b-1\over{b-1}}= 1 \qquad \text{and} \qquad R_2^{(b)}={b^2-1\over{b-1}}= b+1\qquad\text{for}\ b\ge2.

In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as

R_n=R_n^{(10)}={10^n-1\over{10-1}}={10^n-1\over9}\qquad\mbox{for }n\ge1.

Thus, the number Rn = Rn(10) consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base 10 starts with

1, 11, 111, 1111, ... (sequence A002275 in OEIS).

Similarly, the repunits base 2 are defined as

R_n^{(2)}={2^n-1\over{2-1}}={2^n-1}\qquad\mbox{for }n\ge1.

Thus, the number Rn(2) consists of n copies of the digit 1 in base 2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers Mn = 2n − 1.

Properties[edit]

  • Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition. For example,
    R35(b) = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base b in which the repunit is expressed.
  • Any positive multiple of the repunit Rn(b) contains at least n nonzero digits in base b.
  • The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2). The Goormaghtigh conjecture says there are only these two cases.
  • Using the pigeon-hole principle it can be easily shown that for each n and b such that n and b are relatively prime there exists a repunit in base b that is a multiple of n. To see this consider repunits R1(b),...,Rn(b). Assume none of the Rk(b) is divisible by n. Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1≤i<jn such that Ri(b) and Rj(b) have the same residue modulo n. It follows that Rj(b) - Ri(b) has residue 0 modulo n, i.e. is divisible by n. Rj(b) - Ri(b) consists of j - i ones followed by i zeroes. Thus, Rj(b) - Ri(b) = Rj-i(b) x bi . Since n divides the left-hand side it also divides the right-hand side and since n and b are relative prime n must divide Rj-i(b) contradicting the original assumption.
  • The Feit–Thompson conjecture is that Rq(p) never divides Rp(q) for two distinct primes p and q.

Factorization of decimal repunits[edit]

R1 = 1
R2 = 11
R3 = 3 · 37
R4 = 11 · 101
R5 = 41 · 271
R6 = 3 · 7 · 11 · 13 · 37
R7 = 239 · 4649
R8 = 11 · 73 · 101 · 137
R9 = 3 · 3 · 37 · 333667
R10 = 11 · 41 · 271 · 9091
R11 = 21649 · 513239
R12 = 3 · 7 · 11 · 13 · 37 · 101 · 9901
R13 = 53 · 79 · 265371653
R14 = 11 · 239 · 4649 · 909091
R15 = 3 · 31 · 37 · 41 · 271 · 2906161
R16 = 11 · 17 · 73 · 101 · 137 · 5882353
R17 = 2071723 · 5363222357
R18 = 3 · 3 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667
R19 = 1111111111111111111
R20 = 11 · 41 · 101 · 271 · 3541 · 9091 · 27961
R21 = 3 · 37 · 43 · 239 · 1933 · 4649 · 10838689
R22 = 11 · 11 · 23 · 4093 · 8779 · 21649 · 513239
R23 = 11111111111111111111111
R24 = 3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · 99990001
R25 = 41 · 271 · 21401 · 25601 · 182521213001
R26 = 11 · 53 · 79 · 859 · 265371653 · 1058313049
R27 = 3 · 3 · 3 · 37 · 757 · 333667 · 440334654777631
R28 = 11 · 29 · 101 · 239 · 281 · 4649 · 909091 · 121499449
R29 = 3191 · 16763 · 43037 · 62003 · 77843839397
R30 = 3 · 7 · 11 · 13 · 31 · 37 · 41 · 211 · 241 · 271 · 2161 · 9091 · 2906161

Repunit primes[edit]

The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.

It is easy to show that if n is divisible by a, then Rn(b) is divisible by Ra(b):

R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b)

where \Phi_d(x) is the d^\mathrm{th} cyclotomic polynomial and d ranges over the divisors of n. For p prime, \Phi_p(x)=\sum_{i=0}^{p-1}x^i, which has the expected form of a repunit when x is substituted with b.

For example, 9 is divisible by 3, and thus R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials \Phi_3(x) and \Phi_9(x) are x^2+x+1 and x^6+x^3+1 respectively. Thus, for Rn to be prime n must necessarily be prime. But it is not sufficient for n to be prime; for example, R3 = 111 = 3 · 37 is not prime. Except for this case of R3, p can only divide Rn for prime n if p = 2kn + 1 for some k.

Decimal repunit primes[edit]

Rn is prime for n = 2, 19, 23, 317, 1031,... (sequence A004023 in OEIS). R49081 and R86453 are probably prime. On April 3, 2007 Harvey Dubner (who also found R49081) announced that R109297 is a probable prime.[2] He later announced there are no others from R86453 to R200000.[3] On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime,[4] along with his intent to search to 400000. As of November 2012, all further candidates up to R2500000 have been tested, but no new probable primes have been found so far.

It has been conjectured that there are infinitely many repunit primes[5] and they seem to occur roughly as often as the prime number theorem would predict: the exponent of the Nth repunit prime is generally around a fixed multiple of the exponent of the (N-1)th.

The prime repunits are a trivial subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.

Base-2 repunit primes[edit]

Main article: Mersenne prime

Base-2 repunit primes are called Mersenne primes.

Base-3 repunit primes[edit]

The first few base-3 repunit primes are

13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013, ... (sequence A076481 in OEIS),

corresponding to n of

3, 7, 13, 71, 103, ... (sequence A028491 in OEIS).

Base-4 repunit primes[edit]

The only base-4 repunit prime is 5 (11_4). 4^n-1=\left(2^n+1\right)\left(2^n-1\right), and 3 always divides 2^n+1 when n is odd and 2^n-1 when n is even. For n greater than 2, both 2^n+1 and 2^n-1 are greater than 3, so removing the factor of 3 still leaves two factors greater than 1, so the number cannot be prime.

Base 5 repunit primes[edit]

The first few base-5 repunit primes are

31, 19531, 12207031, 305175781, 177635683940025046467781066894531, (sequence A086122 in OEIS)

corresponding to n of

3, 7, 11, 13, 47, ... (sequence A004061 in OEIS).

Base 6 repunit primes[edit]

The first few base-6 repunit primes are

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, ..., (sequence A165210 in OEIS)

corresponding to n of

2, 3, 7, 29, 71, ... (sequence A004062 in OEIS)

Base 7 repunit primes[edit]

The first few base 7 repunit primes are

2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601

corresponding to n of

5, 13, 131, 149, ... (sequence A004063 in OEIS)

Base 8 and 9 repunit primes[edit]

The only base-8 or base-9 repunit prime is 73 (111_8). 8^n-1=\left(4^n+2^n+1\right)\left(2^n-1\right), and 7 divides 4^n+2^n+1 when n is not divisible by 3 and 2^n-1 when n is a multiple of 3. 9^n-1=\left(3^n+1\right)\left(3^n-1\right), and 2 always divides both 3^n+1 and 3^n-1.

Base 12 repunit primes[edit]

The first few base 12 repunit primes are

13, 157, 22621, 29043636306420266077, 435700623537534460534556100566709740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941

corresponding to n of

2, 3, 5, 19, 97, 109, 317, 353, 701, ... (sequence A004064 in OEIS)

Base 20 repunit primes[edit]

The only known vigesimal (base 20) repunit primes or probable primes are for n of

3, 11, 17, 1487, 31013, 48859, 61403 (sequence A127995 in OEIS)

The first three of these in decimal are

421, 10778947368421 and 689852631578947368421

The smallest repunit primes of any natural number base[edit]

(sequence A084740 in OEIS)

Base 1 2 3 4 5 6 7 8 9 10
min n 2 2 3 2 3 2 5 3 None 2
Base 11 12 13 14 15 16 17 18 19 20
min n 17 2 5 3 3 2 3 2 19 3
Base 21 22 23 24 25 26 27 28 29 30
min n 3 2 5 3 None 7 3 2 5 2
Base 31 32 33 34 35 36 37 38 39 40
min n 7 None 3 13 313 2 13 3 349 2
Base 41 42 43 44 45 46 47 48 49 50
min n 3 2 5 5 19 2 127 19 None 3
Base 51 52 53 54 55 56 57 58 59 60
min n 4229 2 11 3 17 7 3 2 3 2
Base 61 62 63 64 65 66 67 68 69 70
min n 7 3 5 None 19 2 19 5 3 2
Base 71 72 73 74 75 76 77 78 79 80
min n 3 2 5 5 3 41 3 2 5 3
Base 81 82 83 84 85 86 87 88 89 90
min n None 2 5 17 5 11 7 2 3 3
Base 91 92 93 94 95 96 97 98 99 100
min n 4421 439 7 5 7 2 17 13 3 2
Base 101 102 103 104 105 106 107 108 109 110
min n 3 2 19 97 3 2 17 2 17 3
Base 111 112 113 114 115 116 117 118 119 120
min n 3 2 23 29 7 59 3 5 3 5
Base 121 122 123 124 125 126 127 128 129 130
min n None 5 43 599 None 2 5 7 5 2
Base 131 132 133 134 135 136 137 138 139 140
min n 3 47 13 5 1171 2 11 2 163 79
Base 141 142 143 144 145 146 147 148 149 150
min n 3 1231 3 None 5 7 3 2 7 3

All bases up to 150 except powers (which can be written as mn, with n > 1) have known proved repunit prime number.

The smallest natural number base b that R_p is prime for prime p[edit]

The list is about the first 100 primes. (sequence A066180 in OEIS)

p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
min b 2 2 2 2 5 2 2 2 10 6 2 61 14 15 5 24 19 2 46 3
p 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
min b 11 22 41 2 12 22 3 2 12 86 2 7 13 11 5 29 56 30 44 60
p 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
min b 304 5 74 118 33 156 46 183 72 606 602 223 115 37 52 104 41 6 338 217
p 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
min b 13 136 220 162 35 10 218 19 26 39 12 22 67 120 195 48 54 463 38 41
p 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
min b 17 808 404 46 76 793 38 28 215 37 236 59 15 514 260 498 6 2 95 3

List of repunit primes base b[edit]

b n which R_n(b) is prime (numbers greater than 232768 are only PRP, these n's are checked up to 100000) OEIS sequence
−30 2, 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599 A071382
−29 7
−28 3, 19, 373, 419, 491, 1031, 83497 A071381
−27 None (Algebra)
−26 11, 109, 227, 277, 347, 857, 2297, 9043 A071380
−25 3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863, 43201, 78707 A057191
−24 2, 7, 11, 19, 2207, 2477, 4951 A057190
−23 11, 13, 67, 109, 331, 587, 24071, 29881, 44053 A057189
−22 3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287 A057188
−21 3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599 A057187
−20 2, 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967 A057186
−19 17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951 A057185
−18 2, 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147 A057184
−17 7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259 A057183
−16 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197 A057182
−15 3, 7, 29, 1091, 2423, 54449, 67489, 551927 A057181
−14 2, 7, 53, 503, 1229, 22637, 1091401 A057180
−13 3, 11, 17, 19, 919, 1151, 2791, 9323, 56333 A057179
−12 2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739 A057178
−11 5, 7, 179, 229, 439, 557, 6113, 223999, 327001 A057177
−10 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207 A001562
−9 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247 A057175
−8 2 (Algebra)
−7 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653 A057173
−6 2, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337 A057172
−5 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429 A057171
−4 2, 3 (Aurifeuillean factorization)
−3 2, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897 A007658
−2 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ......, 13347311, 13372531 (1 is not prime) A000978
−1 None (1 is not prime)
0 None (1 is not prime)
1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, ...... (All primes) A000040
2 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, ......, 32582657, ......, 37156667, ......, 42643801, ......, 43112609, ......, 57885161 A000043
3 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843 A028491
4 2 (Algebra)
5 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413 A004061
6 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099 A004062
7 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699 A004063
8 3 (Algebra)
9 None (Algebra)
10 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343 A004023
11 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831 A005808
12 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889 A004064
13 5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089 A016054
14 3, 7, 19, 31, 41, 2687, 19697, 59693, 67421 A006032
15 3, 43, 73, 487, 2579, 8741, 37441, 89009 A006033
16 2 (Algebra)
17 3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509 A006034
18 2, 25667, 28807, 142031, 157051, 180181 A133857
19 19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359 A006035
20 3, 11, 17, 1487, 31013, 48859, 61403 A127995
21 3, 11, 17, 43, 271, 156217, 328129 A127996
22 2, 5, 79, 101, 359, 857, 4463, 9029, 27823 A127997
23 5, 3181, 61441, 91943 A204940
24 3, 5, 19, 53, 71, 653, 661, 10343, 49307 A127998
25 None (Algebra)
26 7, 43, 347, 12421, 12473, 26717 A127999
27 3 (Algebra)
28 2, 5, 17, 457, 1423 A128000
29 5, 151, 3719, 49211, 77237 A181979
30 2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883 A098438

For more information, see Repunit primes in base -50 to 50, Repunit primes in base 2 to 150, Repunit primes in base -150 to -2, and Repunit primes in base -200 to -2.

Algebra factorization of repunit numbers[edit]

If b is a power (can be written as mn, with m, n integers, n > 1), then there is at most one repunit in base b. If n is a prime power (can be written as pr, with p prime, r integer, p, r >0), then all repunit in base b are not prime aside from R_p, R_p, can be either prime or composite, the former examples, b = -216, -128, 4, 8, 16, 27, 36, 100, 128, 256, etc, the letter examples, b = -243, -125, -64, -32, -27, -8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc. if n is not a prime power, then no base b repunit exists, for example, b = 64 (with n = 6), and b = -1, 0, or 1 (with n can be any natural number). Another special situation is b = -4, which has the aurifeuillean factorization 4^n+1 = 4^{2k-1}+1 = (2^{2k-1}-2^k+1) (2^{2k-1}+2^k+1).There is also a conjecture that when b is neither a power nor -4, then there are infinity many base b repunit primes.

History[edit]

Although they were not then known by that name, repunits in base 10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of recurring decimals.[6]

It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the factorization by such mathematicians as Reuschle of all repunits up to R16 and many larger ones. By 1880, even R17 had been factored[7] and it is curious that, though Édouard Lucas showed no prime below three million had period nineteen, there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician Oscar Hoppe proved R19 to be prime in 1916[8] and Lehmer and Kraitchik independently found R23 to be prime in 1929.

Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected. R317 was found to be a probable prime circa 1966 and was proved prime eleven years later, when R1031 was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes.

Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size.

The Cunningham project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.

Demlo numbers[edit]

The Demlo numbers[9] 1, 121, 12321, 1234321, … 12345678987654321, 1234567900987654321, 123456790120987654321, … were defined by D. R. Kaprekar as the squares of the repunits, resolving the uncertainty how to continue beyond the highest digit (9), and named after Demlo railway station 30 miles from Bombay on the then G.I.P. Railway, where he thought of investigating them.

See also[edit]

Notes[edit]

  1. ^ Beiler, Albert (2013). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains (2 ed.). New York: Dover Publications. p. 83. ISBN 978-0-486-21096-4. 
  2. ^ Harvey Dubner, New Repunit R(109297)
  3. ^ Harvey Dubner, Repunit search limit
  4. ^ Maksym Voznyy, New PRP Repunit R(270343)
  5. ^ Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages.
  6. ^ Dickson, Leonard Eugene and Cresse, G.H.; History of the Theory of Numbers; pp. 164-167 ISBN 0-8218-1934-8
  7. ^ Dickson and Cresse, pp. 164-167
  8. ^ Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.
  9. ^ Weisstein, Eric W., "Demlo Number", MathWorld.

External links[edit]

Web sites[edit]

Books[edit]