Prove that no Fermat number is a $3$ rd power of an integer.

Question

Let $F_n$ be the $n$ th Fermat number, $F_n:=2^{2^n}+1$.

I have been working with a similar question that reads: "Prove that no Fermat number is a perfect square." Where I found an answer that reads: "$F_0 $ and $F_1$ ($3$ and $5$ respectively) are clearly not perfect squares.

For $F_n$ where $n \ge 2$, $F_n \equiv 7 \pmod {10}$. However, only numbers that are congruent to $0, 1, 4, 5, 6$, or $9 \pmod {10}$ can be perfect squares."

I do not know if this is correct and if so how to use the same method to prove the same for a $3$ rd power integer.

Answer 1

Just to clarify, Fermat number is a number of the type $2^{2^n}+1$.

About the part that no Fermat number ($F_n$) is a perfect square, I think the reasoning provided is correct.

$F_n\equiv 7 \pmod {10}$ is true because of the following:

$F_n=2^{2^n}+1=(2^2)^{2^{n-1}}+1=(5-1)^{2^{n-1}}+1\equiv 2\pmod 5$ (Using binomial theorem).

Since $F_n$ is odd, $F_n\equiv 2 \pmod {5}\implies F_n \equiv 7 \pmod{10}$.

And since no square has $7$ as its last digit, $F_n$ can never be a perfect square.

However, cubes do attain every possible last digit so this same logic won't be able to show that $F_n$ is never a perfect cube.

Answer 2

To prove that no Fermat number can be represented as the cube of an integer, start by assuming that it can, and let $2^{2^n} + 1= k^3$, then $2^{2^n}$ can be factored into $(k-1)(k^2+k+1)$ for an integer k.

This means that $k-1 = 2^a$ and $k^2+k+1 = 2^b$ with $a+b = 2^n$

Note that $k^2 + k + 1 \text{ mod 2} = 1$ for integer values of $k$ (why?)

The solution should follow simply by noting that $2^b$ must equal $1$ by the above conclusion, which shows that no values of $k$ satisfy the equality and thus an contradiction arises.

This can be expanded to prove that no Fermat number can be represented as an odd power.

$k^{2m-1} -1 = (k-1)(\sum_{i=0}^{2m-2} k^i)$ and $\sum_{i=0}^{2m-2} k^i \text{ mod 2} = 1$

and so by a similiar argument to the one above, the proposition can be proven.

The quadratic residue proof suffices to prove no Fermat number can be represented as an even power by noting that $k^{2m} = (k^m)^2$.

Therefore, more generally, no Fermat number $2^{2^{N}} + 1$ can be represented as $k^n$ for an integer $k$ with $n \geq 2$