---
tag: 0N9E4G2P97
kind: definition
name: Fermat prime
msc: [11A41]
defines: 06N6JAZF44
uses: [0R03B54V4Z, 0SN0MAEKYX]
sources:
  - url: https://en.wikipedia.org/wiki/Fermat_number
    locator: Primality
---

## Definition

A Fermat prime is a [[Fermat number|0R03B54V4Z]] $F_n = 2^{2^n} + 1$ that is a [[prime number|0SN0MAEKYX]].

## 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.