Fermat prime

Definition

A Fermat prime is a Fermat number $F_n = 2^{2^n} + 1$ that is a prime number.

Remarks

The numbers $F_0, \dots, F_4$, that is $3, 5, 17, 257, 65537$, are prime. No other Fermat prime is known. Fermat conjectured that all Fermat numbers are prime. Euler refuted this in 1732 with $F_5 = 641 \cdot 6700417$. A regular $n$-gon is constructible with ruler and compass if and only if $n$ is a power of $2$ times distinct Fermat primes. This is the Gauss–Wantzel theorem.

Explanations

L1 human

A Fermat prime is a prime of the form $2^{2^n} + 1$. Only five of them are known, and it is an open problem whether there are more.