Distinct Fermat numbers are coprime
= Distinct Fermat numbers are coprime
{title2=$\gcd(F_m,F_n)=1\quad(m\ne n)$}
If $m<n$, a common <divisor> of $F_m$ and $F_n$ divides both $F_n-2$, by the <Fermat number product identity>, and $F_n$. It therefore divides two. Both <Fermat numbers> are <odd integers>, so their <greatest common divisor> is one. Each term's <prime factors> are consequently different from the <prime factors> of every other term.