# 209 (number)

From HandWiki

__: Natural number__

**Short description**
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Cardinal | two hundred nine | |||

Ordinal | 209th (two hundred ninth) | |||

Factorization | 11 × 19 | |||

Greek numeral | ΣΘ´ | |||

Roman numeral | CCIX | |||

Binary | 11010001_{2} | |||

Ternary | 21202_{3} | |||

Quaternary | 3101_{4} | |||

Quinary | 1314_{5} | |||

Senary | 545_{6} | |||

Octal | 321_{8} | |||

Duodecimal | 155_{12} | |||

Hexadecimal | D1_{16} | |||

Vigesimal | A9_{20} | |||

Base 36 | 5T_{36} |

**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]}^{[2]}209 partial permutations on four elements,^{[3]}^{[4]}and 209 distinct undirected simple graphs on 7 or fewer unlabeled vertices.^{[5]}^{[6]} - 209 is the smallest number with six representations as a sum of three positive squares.
^{[7]}These representations are:- 209 = 1
^{2}+ 8^{2}+ 12^{2}= 2^{2}+ 3^{2}+ 14^{2}= 2^{2}+ 6^{2}+ 13^{2}= 3^{2}+ 10^{2}+ 10^{2}= 4^{2}+ 7^{2}+ 12^{2}= 8^{2}+ 8^{2}+ 9^{2}.

- 209 = 1

- 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.
^{[8]}^{[9]}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.^{[10]}

## References

- ↑ Sloane, N. J. A., ed. "Sequence A001353 (a(n) = 4*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1)". OEIS Foundation. https://oeis.org/A001353.
- ↑ Kreweras, Germain (1978), "Complexité et circuits eulériens dans les sommes tensorielles de graphes" (in French),
*Journal of Combinatorial Theory*, Series B**24**(2): 202–212, doi:10.1016/0095-8956(78)90021-7 - ↑ 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)". OEIS Foundation. https://oeis.org/A002720.
- ↑ Laradji, A.; Umar, A. (2007), "Combinatorial results for the symmetric inverse semigroup",
*Semigroup Forum***75**(1): 221–236, doi:10.1007/s00233-007-0732-8 - ↑ Sloane, N. J. A., ed. "Sequence A006897 (Hierarchical linear models on n factors allowing 2-way interactions; or graphs with <= n nodes.)". OEIS Foundation. https://oeis.org/A006897.
- ↑ Adams, Peter; Eggleton, Roger B.; MacDougall, James A. (2006), "Taxonomy of graphs of order 10",
*Congressus Numerantium***180**: 65–80, https://people.smp.uq.edu.au/PeterAdams/research/poset10/graphsorder10.pdf - ↑ 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.)". OEIS Foundation. https://oeis.org/A025414.
- ↑ Sloane, N. J. A., ed. "Sequence A057588 (Kummer numbers: -1 + product of first n consecutive primes)". OEIS Foundation. https://oeis.org/A057588.
- ↑ O'Shea, Owen (2016),
*The Call of the Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics*, Prometheus Books, p. 44, ISBN 9781633881488, https://books.google.com/books?id=xgIdDgAAQBAJ&pg=PA44 - ↑ Sloane, N. J. A., ed. "Sequence A125549 (Composite Kummer numbers)". OEIS Foundation. https://oeis.org/A125549.

Original source: https://en.wikipedia.org/wiki/209 (number).
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