For all odd primes, $p^2 \equiv 1 \pmod 8$. Via Dirichlet, we know there are an infinite number of primes of the form $q \equiv 7 \pmod 8$. Therefore there should be at least some primes $q = p^2 – 2$.
However, it's obvious that not all positive integers $n \equiv 7 \pmod 8$ are prime, and the overlap of those that are prime with integers of the form $p^2 – 2$ isn't obviously infinite.
Is there proof that there are/are not an infinite number of primes of this form?
Best Answer
We have no idea. We don't even know if there are infinitely many primes of the form $X^2 - 2$. In fact, we don't have a single example of a quadratic polynomial that we can prove takes infinitely many prime values. This is far beyond current scope.
This is closely related to Schinzel's Hypothesis H.