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For example, $(3,5), (5,13)$ or $(11,61)$ satisfies this condition, but I have no idea how to prove or disprove that such pairs exist infinitely many. Is there any prior researches about it?

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    $\begingroup$ Interesting that someone (anonymously) voted to close this query. Given the responses (i.e. answers) that this query has received, what was the OP supposed to do - show work on what is actually an unsolved problem? $\endgroup$ Feb 23, 2021 at 21:13
  • $\begingroup$ OP should be able to prove the Riemann Hypothesis if he wants to be in MSE @user2661923 $\endgroup$
    – Some Guy
    Feb 23, 2021 at 21:17

2 Answers 2

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OEIS/A048161 says

A048161: Primes $p$ such that $q = (p^2 + 1)/2$ is also a prime.

Primes which are a leg of an integral right triangle whose hypotenuse is also prime.

It is conjectured that there are an infinite number of such triangles.

So, no one knows whether infinitely many such pairs exist.

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  • $\begingroup$ I didn't know that the same conjecture already exists, even though I felt I have searched enough! Thank you for finding such a clear and simple evidence. $\endgroup$
    – okw1124
    Mar 3, 2021 at 0:34
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This is covered by the generalized Bunyakovsky conjecture (aka generalized Dickson conjecture), for the polynomials $f_1(x) = 2x+1$ and $f_2(x) = 2 x^2 + 2 x + 1$. However, the only case where this conjecture is known to be true is for just one polynomial of degree $1$ (Dirichlet's theorem).

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  • $\begingroup$ I'm surprised that someone had same questions more than a hundred years ago. Thank you for giving me an advice. $\endgroup$
    – okw1124
    Mar 3, 2021 at 2:43

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