[Math] Infinitely many prime numbers of the form $n^{2^k}+1$

Question

I am not a specialist of number theory, so please excuse my ignorance: is the following question still an open problem?
Let $k \in \mathbb{N}^*$, are there infinitely many prime numbers of the form $n^{2^k}+1$?

Answer 1

Your question is still open. It is a special case of Schinzel's Hypothesis H applied to the polynomial $f(x)=x^{2^k}+1$.

As Bjorn mentions in his comment, the case of $k=1$ is a particularly famous unsolved problem. It is the fourth of Landau's problems (Edmund Landau was a famous German number theorist during the early twentieth century).