Fermat number product identity
= Fermat number product identity
{c}
{title2=$\prod_{j=0}^{n-1}F_j=F_n-2$}
For each <positive integer> $n$, the product of the preceding <Fermat numbers> is two less than the next:
$$
\prod_{j=0}^{n-1}F_j=2^{2^n}-1=F_n-2.
$$
For example, $3\cdot5\cdot17=255=257-2$. 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.