Fermat Number is not Cube

Theorem

There exist no Fermat numbers which are cubes.

$\blacksquare$

Let $1 + 2^n$ be a Fermat number.

Suppose $a^3 = 1 + 2^n$.

Since $1 + 2^n$ is odd, $a$ must also be odd.

Write $a = 2 m + 1$, where $m > 0$.

Then:

$\ds 1 + 2^n$

$=$

$\ds \paren {2 m + 1}^3$

$\ds $

$=$

$\ds 8 m^3 + 12 m^2 + 6 m + 1$

$\ds 2^n$

$=$

$\ds 8 m^3 + 12 m^2 + 6 m$

$\ds $

$=$

$\ds 2 m \paren {4 m^2 + 6 m + 3}$

Since $4 m^2 + 6 m + 3$ is odd and greater than $1$, it is not a power of $2$.

Hence the equation $a^3 = 1 + 2^n$ has no solution in the integers.

Thus there are no Fermat numbers which are cubes.

$\blacksquare$