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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?

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    $\begingroup$ Given how little is known about similar problems, I doubt much is known here. $\endgroup$ Apr 10, 2021 at 4:30
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    $\begingroup$ oeis.org/A062326 is the sequence of $p$s that work. $\endgroup$ Apr 10, 2021 at 4:48

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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.

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  • $\begingroup$ 1 or 7 mod 8 divisors by quadratic reciprocity if they exist ... $\endgroup$ Apr 17, 2021 at 23:20

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