209 (number)

209 (two hundred [and] nine) is the natural number following 208 and preceding 210.

← 208 209 210 →
← 200 201 202 203 204 205 206 207 208 209 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
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

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 = 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.^[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

1. 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.
2. Kreweras, Germain (1978), "Complexité et circuits eulériens dans les sommes tensorielles de graphes" [Complexity & Eulerian circuits in graphic tensorial sums], Journal of Combinatorial Theory, Series B (in French), 24 (2): 202–212, doi:10.1016/0095-8956(78)90021-7, MR 0486144
3. 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.
4. 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, MR 2351933, S2CID 122239867
5. Sloane, N. J. A. (ed.). "Sequence A006897 (Hierarchical linear models on n factors allowing 2-way interactions; or graphs with <= n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Adams, Peter; Eggleton, Roger B.; MacDougall, James A. (2006), "Taxonomy of graphs of order 10" (PDF), Proceedings of the Thirty-Seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, 180: 65–80, MR 2311249
7. 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.
8. 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.
9. O'Shea, Owen (2016), The Call of the Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics, Prometheus Books, p. 44, ISBN 9781633881488
10. Sloane, N. J. A. (ed.). "Sequence A125549 (Composite Kummer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.