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

From Product of Sequence of Fermat Numbers plus 2:

\(\ds \forall n \in \Z_{>0}: \, \)

\(\ds F_{n + m}\)

\(=\)

\(\ds \prod_{j \mathop = 0}^{n + m - 1} F_j + 2\)

\(\ds \leadsto \ \ \)

\(\ds F_{n + m} - 2\)

\(=\)

\(\ds \prod_{j \mathop = 0}^{n + m - 1} F_j\)

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

This of course includes $F_n$.

$\blacksquare$