# 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.
Mentioned in ?
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.
For g = 2 (real quadratic case), exceptional prime p is only the Fermat primes and these are known only 3, 5, 17, 257, 65537.
For real quadratic fields, the exceptional primes are Fermat primes.
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.
There are so many different kinds of primes (Sophie Germain primes, Fermat primes, Mersenne primes, factorial primes, primorial primes, etc.

Site: Follow: Share:
Open / Close