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:
\(\ds \forall n \in \Z_{>0}: \, \) | \(\ds F_{n + m}\) | \(=\) | \(\ds \prod_{j \mathop = 0}^{n + m - 1} F_j + 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds F_{n + m} - 2\) | \(=\) | \(\ds \prod_{j \mathop = 0}^{n + m - 1} F_j\) |
and so all Fermat numbers of index less than $n + m$ are divisors of $F_{n + m} - 2$.
This of course includes $F_n$.
$\blacksquare$