# Goldbach's Theorem

## Contents

## Theorem

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

Then $F_m$ and $F_n$ are coprime.

## Proof 1

Aiming for a contradiction, suppose $F_m$ and $F_n$ have a common divisor $p$ which is prime.

As both $F_n$ and $F_m$ are odd, it follows that $p$ must itself be odd.

Without loss of generality, suppose that $m > n$.

Then $m = n + k$ for some $k \in \Z_{>0}$.

\(\displaystyle F_m - 1\) | \(\equiv\) | \(\displaystyle -1\) | \(\displaystyle \pmod p\) | as $p \divides F_m$ | |||||||||

\(\displaystyle F_n - 1\) | \(\equiv\) | \(\displaystyle -1\) | \(\displaystyle \pmod p\) | as $p \divides F_n$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {F_n - 1}^{2^k}\) | \(\equiv\) | \(\displaystyle -1\) | \(\displaystyle \pmod p\) | Fermat Number whose Index is Sum of Integers | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {-1}^{2^k}\) | \(\equiv\) | \(\displaystyle -1\) | \(\displaystyle \pmod p\) | Congruence of Product | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1\) | \(\equiv\) | \(\displaystyle -1\) | \(\displaystyle \pmod p\) | Congruence of Powers | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(\equiv\) | \(\displaystyle 2\) | \(\displaystyle \pmod p\) |

Hence $p = 2$.

However, it has already been established that $p$ is odd.

From this contradiction it is deduced that there is no such $p$.

Hence the result.

$\blacksquare$

## Proof 2

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

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

From the corollary to Product of Sequence of Fermat Numbers plus 2:

- $F_m \divides F_n - 2$

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.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $9$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $257$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Fermat number**(P. de Fermat, 1640)

- 2001: Michal Křížek, Florian Luca and Lawrence Somer:
*17 Lectures on Fermat Numbers*: Theorem $4.1$