Fermat number product identity (${\prod }_{j=0}^{n-1}{F}_j=F_n-2$)

For each positive integer $n$, the product of the preceding Fermat numbers is two less than the next:For example, $3\cdot 5\cdot 17=255=257-2$. The identity telescopes by repeated difference of squares: $(2^{2^j}-1)(2^{2^j}+1)=2^{2^{j+1}}-1$. Its usefulness is that a huge integer product exposes a tiny difference, two, from the next term.

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.