Fermat prime
A prime number of the form 2^2^n + 1. Any prime number of the form 2^n+1 must be a Fermat prime. Fermat conjectured in a letter to someone or other that all numbers 2^2^n+1 are prime, having noticed that this is true for n=0,1,2,3,4. Euler proved that 641 is a factor of 2^2^5+1. Of course nowadays we would just ask a computer, but at the time it was an impressive achievement (and his proof is very elegant). No further Fermat primes are known several have been factorised, and several more have been proved composite without finding explicit factorisations. Gauss proved that a regular N-sided polygon can be constructed with ruler and compasses if and only if N is a power of 2 times a product of distinct Fermat primes. (1995-04-10) In addition suitable contents:
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