# Goldbach's Theorem/Proof 2

## Theorem

Let $F_m$ and $F_n$ be Fermat numbers such that $m \ne n$.

Then $F_m$ and $F_n$ are coprime.

## Proof

Let $F_m$ and $F_n$ be Fermat numbers such that $m < n$.

Let $d = \gcd \set {F_m, F_n}$.

$F_m \divides F_n - 2$

But then:

 $\displaystyle d$ $\divides$ $\displaystyle F_n$ Definition of Common Divisor of Integers $\, \displaystyle \land \,$ $\displaystyle d$ $\divides$ $\displaystyle F_m$ (where $\divides$ denotes divisibility) $\displaystyle \leadsto \ \$ $\displaystyle d$ $\divides$ $\displaystyle F_n - 2$ as $F_m \divides F_n - 2$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $\divides$ $\displaystyle F_n - \paren {F_n - 2}$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $\divides$ $\displaystyle 2$

But all Fermat numbers are odd, so:

$d \ne 2$

It follows that $d = 1$.

The result follows by definition of coprime.

$\blacksquare$

## Source of Name

This entry was named for Christian Goldbach.