---
tag: 05598J0Y8R
kind: statement
role: theorem
name: Fermat numbers are pairwise coprime
aliases: [goldbach-theorem-fermat-numbers]
msc: [11A41]
uses: [0R03B54V4Z, 0GFKP8F8AG]
sources:
  - url: https://en.wikipedia.org/wiki/Fermat_number
    locator: Basic properties
---

## Statement

Let $m$ and $n$ be distinct natural numbers.
Then the [[Fermat numbers|0R03B54V4Z]] $F_m$ and $F_n$ are [[coprime|0GFKP8F8AG]].

## Remarks

The proof uses $F_0 F_1 \cdots F_{n-1} = F_n - 2$. A common divisor of $F_m$ and $F_n$ with $m < n$ divides $2$. Fermat numbers are odd, so the divisor is $1$.
The result is often called Goldbach's theorem, after Christian Goldbach.
A consequence is a new proof that there are [[infinitely many primes|0292EPPPT0]]: each $F_n$ has a prime factor, and these primes are all distinct.

## Explanations

### L1 human

No two Fermat numbers share a prime factor.
For example, $F_2 = 17$ and $F_3 = 257$ are coprime, and $F_5 = 641 \cdot 6700417$ has no factor in common with $3$, $5$, $17$, $257$ or $65537$.