The projective-prime infinitude conjecture

Let $n\ge 2$ be an integer and let $q\ge 2$ be a prime power. Define the projective degree

$$m=\frac{q^n-1}{q-1}=1+q+q^2+\cdots+q^{n-1}.$$

A projective prime is a prime value of $m$. The projective-prime infinitude conjecture. There are infinitely many projective primes. Heuristic arguments and computational evidence support this conjecture, but the infinitude of projective primes remains open; it includes the unresolved infinitude questions for Fermat and Mersenne primes.