Divisor of Fermat Number

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Theorem

Let $F_n$ be a Fermat number.

Let $m$ be divisor of $F_n$.

Then $m$ is in the form:

$k \, 2^{n + 2} + 1$

where $k \in \Z_{>0}$ is an integer.


Proof


Historical Note

In $1747$, Leonhard Paul Euler proved that a divisor of a Fermat number $F_n$ is always in the form $k \, 2^{n + 1} + 1$.

This was later refined to $k \, 2^{n + 2} + 1$, the result given here, by Édouard Lucas.


Sources