Question 23
Find all integers $m,n\ge2$ satisfying both conditions:
(i) $m+1$ is a prime of the form $4k+3$ for an integer $k$.
(ii) There exist a prime $p$ and a nonnegative integer $a$ such that
$$\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.$$
(i) $m+1$ is a prime of the form $4k+3$ for an integer $k$.
(ii) There exist a prime $p$ and a nonnegative integer $a$ such that
$$\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.$$