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*209 is the smallest number with six representations as a sum of three positive squares.<ref>{{SloanesRef|A025414|name=a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.}}</ref> These representations are: |
*209 is the smallest number with six representations as a sum of three positive squares.<ref>{{SloanesRef|A025414|name=a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.}}</ref> These representations are: |
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*:209 {{nowrap|1== 1<sup>2</sup> + 8<sup>2</sup> + 12<sup>2</sup>}} {{nowrap|1== 2<sup>2</sup> + 3<sup>2</sup> + 14<sup>2</sup>}} {{nowrap|1== 2<sup>2</sup> + 6<sup>2</sup> + 13<sup>2</sup>}} {{nowrap|1== 3<sup>2</sup> + 10<sup>2</sup> + 10<sup>2</sup>}} {{nowrap|1== 4<sup>2</sup> + 7<sup>2</sup> + 12<sup>2</sup>}} {{nowrap|1== 8<sup>2</sup> + 8<sup>2</sup> + 9<sup>2</sup>}}. |
*:209 {{nowrap|1== 1<sup>2</sup> + 8<sup>2</sup> + 12<sup>2</sup>}} {{nowrap|1== 2<sup>2</sup> + 3<sup>2</sup> + 14<sup>2</sup>}} {{nowrap|1== 2<sup>2</sup> + 6<sup>2</sup> + 13<sup>2</sup>}} {{nowrap|1== 3<sup>2</sup> + 10<sup>2</sup> + 10<sup>2</sup>}} {{nowrap|1== 4<sup>2</sup> + 7<sup>2</sup> + 12<sup>2</sup>}} {{nowrap|1== 8<sup>2</sup> + 8<sup>2</sup> + 9<sup>2</sup>}}. |
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:By [[Legendre's three-square theorem]], all numbers congruent to 1, 2, 3, 5, or 6 mod 8 have representations as sums of three squares, but this theorem does not explain the high number of such representations for 209. |
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*{{nowrap|1=209 = 2 × 3 × 5 × 7 − 1}}, one less than the product of the first four prime numbers. Therefore 209 is a [[Euclid number]] of the second kind, also called a Kummer number.<ref>{{SloanesRef|A057588|name=Kummer numbers: -1 + product of first n consecutive primes}}</ref> One standard proof of [[Euclid's theorem]] that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer number. However, the Kummer numbers are not all prime, and as a [[semiprime]] (the product of two smaller prime numbers {{nowrap|1=11 × 19}}), 209 is the first example of a composite Kummer number.<ref>{{SloanesRef|A125549|name=Composite Kummer numbers}}</ref> |
*{{nowrap|1=209 = 2 × 3 × 5 × 7 − 1}}, one less than the product of the first four prime numbers. Therefore 209 is a [[Euclid number]] of the second kind, also called a Kummer number.<ref>{{SloanesRef|A057588|name=Kummer numbers: -1 + product of first n consecutive primes}}</ref> One standard proof of [[Euclid's theorem]] that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer number. However, the Kummer numbers are not all prime, and as a [[semiprime]] (the product of two smaller prime numbers {{nowrap|1=11 × 19}}), 209 is the first example of a composite Kummer number.<ref>{{SloanesRef|A125549|name=Composite Kummer numbers}}</ref> |
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*209 can be written as a sum of consecutive integers raised to powers that are consecutive integers:<ref>{{SloanesRef|A003101}}</ref> {{nowrap|1= 209 = 1<sup>6</sup> + 2<sup>5</sup> + 3<sup>4</sup> + 4<sup>3</sup> + 5<sup>2</sup> + 6<sup>1</sup>.}}{{importance-inline}} |
*209 can be written as a sum of consecutive integers raised to powers that are consecutive integers:<ref>{{SloanesRef|A003101}}</ref> {{nowrap|1= 209 = 1<sup>6</sup> + 2<sup>5</sup> + 3<sup>4</sup> + 4<sup>3</sup> + 5<sup>2</sup> + 6<sup>1</sup>.}}{{importance-inline}} |
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Revision as of 18:40, 5 January 2018
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|---|---|---|---|---|
| Cardinal | two hundred nine | |||
| Ordinal | 209th (two hundred ninth) | |||
| Factorization | 11 × 19 | |||
| Greek numeral | ΣΘ´ | |||
| Roman numeral | CCIX | |||
| Binary | 110100012 | |||
| Ternary | 212023 | |||
| Senary | 5456 | |||
| Octal | 3218 | |||
| Duodecimal | 15512 | |||
| Hexadecimal | D116 | |||
209 (two hundred [and] nine) is the natural number following 208 and preceding 210.
In mathematics
- There are 209 spanning trees in a 2 × 5 grid graph,[1] and 209 partial permutations on four elements.[2]
- 209 is the smallest number with six representations as a sum of three positive squares.[3] These representations are:
- 209 = 12 + 82 + 122 = 22 + 32 + 142 = 22 + 62 + 132 = 32 + 102 + 102 = 42 + 72 + 122 = 82 + 82 + 92.
- By Legendre's three-square theorem, all numbers congruent to 1, 2, 3, 5, or 6 mod 8 have representations as sums of three squares, but this theorem does not explain the high number of such representations for 209.
- 209 = 2 × 3 × 5 × 7 − 1, one less than the product of the first four prime numbers. Therefore 209 is a Euclid number of the second kind, also called a Kummer number.[4] One standard proof of Euclid's theorem that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer number. However, the Kummer numbers are not all prime, and as a semiprime (the product of two smaller prime numbers 11 × 19), 209 is the first example of a composite Kummer number.[5]
- 209 can be written as a sum of consecutive integers raised to powers that are consecutive integers:[6] 209 = 16 + 25 + 34 + 43 + 52 + 61.[importance?]
- 209 is a Harshad number (a number that is divisible by the sum of its own digits),[7] because 209/(2+0+9) = 19.[importance?]
See also
- Asteroid 209 Dido
- List of highways numbered 209
- Type 209 submarine, German submarine developed for export
References
- ^ Sloane, N. J. A. (ed.). "Sequence A001353 (a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A002720 (Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A025414 (a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A057588 (Kummer numbers: -1 + product of first n consecutive primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A125549 (Composite Kummer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A003101". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Harshad numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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