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. 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.
^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, MR0486144
^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, MR2311249