What's known about primes of the form $m^n+1$?

Question

For example, the Fermat primes are primes of the form $2^{2^n}+1$.

I'm wondering if the primes $m^n+1$ have a name. More importantly, I'm wondering if there are tables of these primes, and what else is known about them. I'm hoping that someone can point me to a study of these primes, or give any information about them.

Primes $m^n+1$ for $m>1$ and $n>1$ have the not particularly creative name ___generalized Fermat primes___. — Jeppe Stig Nielsen, May 28 '16 at 08:34

score 3 · Answer 1 · answered Sep 03 '13 at 16:53

The first observation, you can make is that if $n$ has and odd factor $d$, then $m^n+1=(m^{n/d}+1)A$ cannot be prime in general. So $n=2^k$ and we arrive at the question of the form $m^{2^k}+1$ with $m=2$ being special case (Fermat's prime). Since almost all questions about Fermat's primes are open (in particular, whether there exists infinite number of them), you can expect the same for your question.

The problem is even harder that one may think. Indeed, if $k=1$ then the question whether $m^2+1$ can be prime infinitely often is also open.

Note that $m$ must be even for $n\gt 0$ – Mark Bennet Sep 03 '13 at 18:07 — Mark Bennet, Sep 03 '13 at 18:07

What's known about primes of the form $m^n+1$?

1 Answers1