641

641 is a natural number that is odd and prime. It is notable in number theory for being a prime factor of the Fermat number F5, which is 2^32 + 1. Specifically, 2^32 + 1 equals 4294967297, and this number factors as 641 × 6700417. Euler showed this divisibility in 1732, providing a counterexample to Fermat's conjecture that all Fermat numbers are prime. The prime 641 can be written in the form 5×2^7 + 1, and indeed 2^32 ≡ -1 (mod 641), so 2^64 ≡ 1 (mod 641). Among Fermat numbers, 641 is the smallest known prime divisor of F5. 641's role in this classic example has made it a standard reference in discussions of primality testing and the history of number theory. See also Fermat number and Fermat primes.