An accessible tour of Fermat numbers F_n = 2^{2^n} + 1 (A000215): from Fermat’s prime guesses to Euler’s disproof with F5 = 4294967297 = 641 × 6700417. We discuss why any prime factor must satisfy special congruence conditions, the pairwise coprimality of Fermat numbers, and the big open questions about infinitely many Fermat primes and related arithmetic curiosities.

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