Definition:Fermat Prime

Definition

A Fermat prime is a Fermat number, i.e. a number of the form $2^{\left({2^n}\right)} + 1$, which happens to be prime.

In fact, $2^{\left({2^n}\right)} + 1$ is prime for $n = 0, 1, 2, 3, 4$.

However, $2^{\left({2^5}\right)} + 1 = 2^{32} + 1$ is divisible by $641$, as was proved by Euler.

No Fermat primes for $n > 4$ have ever been discovered.

Examples

The only known examples of Fermat primes are as follows:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2^{\left({2^0}\right)} + 1\)	\(=\)	\(\displaystyle 3\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2^{\left({2^1}\right)} + 1\)	\(=\)	\(\displaystyle 5\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2^{\left({2^2}\right)} + 1\)	\(=\)	\(\displaystyle 17\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2^{\left({2^3}\right)} + 1\)	\(=\)	\(\displaystyle 257\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2^{\left({2^4}\right)} + 1\)	\(=\)	\(\displaystyle 65\ 537\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This sequence is A019434 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Source of Name

This entry was named for Pierre de Fermat.

He (incorrectly) conjectured that all numbers of the form $2^{\left({2^n}\right)} + 1$ are prime.