Is this expression always irrational?

Question

Is it right that

$$\sqrt[a]{2^{2^n}+1}$$

for every $$a>1,n \in \mathbb N $$

is always irrational?

Answer 1

$$a=1\to 2^{2^n}\in \Bbb Q$$ $$a=2\to \sqrt{2^{2^n}} \in \Bbb Q$$

Answer 2

$\sqrt[a]{m}$ is rational iff it is an integer iff $m$ is an $a$-th power.

The question deals with Fermat numbers.

Apart from the easy counterexamples, Wikipedia says that even a weaker version of the question is an open problem:

Does a Fermat number exist that is not square-free?

Answer 3

As lhf notes, we just have to see that $2^{2^n}+1$ can never be a perfect power. However, since $2^{2^n}$ is a perfect power, using Mihailescu's theorem the only pair of perfect powers differing by $1$ is $8$ and $9$, but $2^{2^n}$ can't be equal to $8$ for $n$ natural. So your expression is, indeed, always irrational.

(I guess I should note that using full Mihailescu's theorem here might be an overkill - 2 being a prime should let us give a relatively simple argument... but hey, whatever works :) )