$n^\pi$ and $n^e$ for positive integer $n\ge2$

Question

Is it known if $n^\pi$ or $n^e$ are integer for some integer $n\ge 2$?

Gelfond - Schneider's theorem does not answer the question because both base and exponent should be algebraic and haven't found nothing about this in internet.

I don't expect a proof, it seems extremely difficult. Only references, if any.

Answer 1

I think Schanuel's conjecture should imply that $n^e$ is never an integer. Here is the idea of how that could go:

Suppose $n^e = m$ then $e \log(n) = \log(m)$.

This would imply $\mathbb{Q}(1, \log(n), \log(m), e, n, m)$ has transcendence degree at most 2, violating the conjecture (assuming we rule out the case that $1$, $\log(n)$ and $\log(m)$ are linearly dependent over $\mathbb{Q}$, but that should be relatively easy)

I haven't thought enough about the $n^\pi$ case, but my guess is you could do something similar.

Of course Schanuel's conjecture is still open, but at least that gives you a place to start looking.