# 209 (number)

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

 ← 208 209 210 →
Cardinaltwo hundred nine
Ordinal209th
(two hundred ninth)
Factorization11 × 19
Greek numeralΣΘ´
Roman numeralCCIX
Binary110100012
Ternary212023
Senary5456
Octal3218
Duodecimal15512

## In mathematics

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. ^
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. ^
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. ^
8. ^
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.