Review question

# Are any two Fermat numbers relatively prime? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8677

## Question

The Fermat numbers $F_n$ are defined by $F_n=2^{2^n}+1$, $n\in\mathbb{Z}^+$.

Show that, for any positive integer $k$, $F_m$ divides $F_{m+k}-2$, and deduce that any two distinct Fermat numbers are relatively prime.

Deduce that there are at least $n+1$ distinct prime numbers less than or equal to $F_n$, for any $n$, and hence that there are infinitely many prime numbers.