# Fermat prime

## Fermat prime

(mathematics)
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.
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References in periodicals archive ?
Almost all number theorists consider the first Fermat prime to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that the first five Fermat numbers are prime.
However no-one has yet proved that [F.sub.4] is the largest Fermat prime.
Let F = Q([square root of 44]), p = 7.(For real quadratic fields, the exceptional primes are Fermat primes.) We note that p|[w.sub.F][[zeta].sub.F](-1), i.e., this case is an exceptional case of Naito [13].
[1] Ikorong Anouk Gilbert Nemron, An original symposium over the Goldbach conjecture, The Fermat primes, The Fermat composite numbers conjecture, and the Mersenne primes conjecture, Mathematicae Notae., XLV(2008), 31-39.
[4] Ikorong Anouk Gilbert Nemron, Runing With The Twin Primes, The Goldbach Conjecture, The Fermat Primes Numbers, The Fermat Composite Numbers, And The Mersenne Primes, Far East Journal Of Mathematical Sciences, 40(2010), 253-266.
[p.sub.t], where the [p'.sub.i]s are distinct Fermat primes.
There are so many different kinds of primes (Sophie Germain primes, Fermat primes, Mersenne primes, factorial primes, primorial primes, etc.), and so many different questions about them.

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