Fermat prime

A Fermat prime is a Fermat number that is also a prime number . A Fermat number F _n is of the form 2 ^m + 1, where m is the n th power of 2 (that is, m = 2 ⁿ , where n is an integer ). To find the Fermat number F _n for an integer n , you first find m = 2 ⁿ , and then calculate 2 ^m + 1. The term arises from the name of a 17th-Century French lawyer and mathematician, Pierre de Fermat, who first defined these numbers and noticed their significance.

Fermat believed that all numbers of the above form are prime numbers; that is, that F _n is prime for all integral values of n . This is indeed the case for n = 0, n = 1, n = 2, n = 3, and n = 4:

When n = 0, m = 2 = 1; therefore
F = 2 ¹ + 1 = 2 + 1 = 3, which is prime

When n = 1,? m = 2 ¹ = 2; therefore
F ₁ = 2 ² + 1 = 4 + 1 = 5, which is prime

When n = 2, m = 2 ² = 4; therefore
F ₂ = 2 ⁴ + 1 = 16 + 1 = 17, which is prime

When n = 3, m = 2 ³ = 8; therefore
F ₃ = 2 ⁸ + 1 = 256 + 1 = 257, which is prime

When n = 4, m = 2 ⁴ = 16; therefore
F ₄ = 2 ¹⁶ + 1 = 65536 + 1 = 65537, which is prime

Using computers, mathematicians have not yet found any Fermat primes for n greater than 4. So far, Fermat's original hypothesis seems to have been wrong. The search continues for Fermat numbers F _n that are prime when n is greater than 4.

Compare Mersenne prime .