Product of Sequence of Fermat Numbers plus 2/Corollary

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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:


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

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

This of course inclues $F_n$.

$\blacksquare$