A Fermat prime is a Fermat number that is also a prime number . A Fermat number *F _{n}* is of the form 2

*+ 1, where*

^{m}*m*is the

*n*th power of 2 (that is,

*m*= 2

^{n}, where

*n*is an integer ). To find the Fermat number

*F*for an integer

_{n}*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 .

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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