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# 209 (number)

 ← 208 209 210 →
Cardinal two hundred nine
Ordinal 209th
(two hundred ninth)
Factorization 11 × 19
Greek numeral ΣΘ´
Roman numeral CCIX
Binary 110100012
Ternary 212023
Quaternary 31014
Quinary 13145
Senary 5456
Octal 3218
Duodecimal 15512
Vigesimal A920
Base 36 5T36

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

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## 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", Journal of Combinatorial Theory, Series B, 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
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.