---
tag: 0R03B54V4Z
kind: definition
name: Fermat number
msc: [11A41]
defines: 0WKG15TV3G
sources:
  - url: https://en.wikipedia.org/wiki/Fermat_number
    locator: lead section
---

## Definition

For a natural number $n$, the $n$-th Fermat number is $F_n = 2^{2^n} + 1$.

## Remarks

The first values are $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ and $F_4 = 65537$.
The Fermat numbers satisfy $F_0 F_1 \cdots F_{n-1} = F_n - 2$ for $n \ge 1$.
A Fermat number that is prime is a [[Fermat prime|0N9E4G2P97]].

## Explanations

### L1 human

The Fermat numbers grow very fast, because the exponent itself is a power of $2$.
For example, $F_5 = 2^{32} + 1 = 4294967297$.