# Product of Sequence of Fermat Numbers plus 2/Corollary

## Corollary to Product of Sequence of Fermat Numbers plus 2

Let $F_n$ denote the $n$th Fermat number.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Then:

$F_n \divides F_{n + m} - 2$

where $\divides$ denotes divisibility.

## Proof

 $\displaystyle \forall n \in \Z_{>0}: \ \$ $\displaystyle F_{n + m}$ $=$ $\displaystyle \prod_{j \mathop = 0}^{n + m - 1} F_j + 2$ $\displaystyle \leadsto \ \$ $\displaystyle F_{n + m} - 2$ $=$ $\displaystyle \prod_{j \mathop = 0}^{n + m - 1} F_j$

and so all Fermat numbers of index less than $n + m$ are divisors of $F_{n + m} - 2$.

This of course inclues $F_n$.

$\blacksquare$