A Fermat number is \(F_k = 2^{2^k} + 1\): the sequence runs 3, 5, 17, 257, 65537, 4294967297.
Fermat conjectured in 1640 that all of them are prime. He was wrong in the most instructive way possible: \(F_0\) through \(F_4\) are prime, but Euler factored \(F_5 = 4294967297 = 641 × 6700417\) in 1732, and no Fermat prime beyond \(F_4 = 65537\) has ever been found despite checking dozens more.
The five known Fermat primes have a beautiful geometric meaning (Gauss–Wantzel): a regular polygon with \(n\) sides is constructible with compass and straightedge exactly when \(n\) is a power of 2 times a product of distinct Fermat primes — which is why the regular 17-gon (Gauss's youthful triumph) and 65537-gon are constructible. 65537 is also the standard RSA public exponent.