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A prime number of the form 2^2^n + 1. Anyprime number of the form 2^n+1 must be a Fermat prime.Fermat conjectured in a letter to someone or other that allnumbers 2^2^n+1 are prime, having noticed that this is truefor n=0,1,2,3,4.Euler proved that 641 is a factor of 2^2^5+1. Of coursenowadays we would just ask a computer, but at the time it wasan impressive achievement (and his proof is very elegant).No further Fermat primes are known; several have beenfactorised, and several more have been proved compositewithout finding explicit factorisations.Gauss proved that a regular N-sided polygon can beconstructed with ruler and compasses if and only if N is apower of 2 times a product of distinct Fermat primes.