Divisor of Fermat Number

Theorem

Let $F_n$ be a Fermat number.

Let $m$ be divisor of $F_n$.

Result by Euler

Then $m$ is in the form:

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

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

Refinement by Lucas

Let $n \ge 2$.

Then $m$ is in the form:

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

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 $k \, 2^{n + 2} + 1$ by François Édouard Anatole Lucas.