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Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect square. Since prime numbers have zero density, this theorem does not apply to them. And I was wondering if there is a result in this direction regarding prime numbers. Explicitily - are there infinitely many pairs of prime numbers $(p,q)$ such that $p-q$ is a perfect square

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    $\begingroup$ Almost assuredly, and that square might be less than 246. Gerhard "See About Small Prime Gaps" Paseman, 2018.10.02. $\endgroup$ – Gerhard Paseman Oct 2 '18 at 17:01
  • $\begingroup$ Well, obvious, thank you, I will defend myself with saying that I am sleepy. However it is not proved for sure, as you say, so the question still has something actual to ask (with the problem most probably being quite easier). $\endgroup$ – Stoyan Apostolov Oct 2 '18 at 17:45
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    $\begingroup$ I think, should such a result be established, that this square can be $ 4 $. Googling "jumping champions" may be insightful. $\endgroup$ – Sylvain JULIEN Oct 2 '18 at 18:08
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Tao and Ziegler extended the Green-Tao theorem to the polynomial setting. As a very special case we get that any subset of the primes with positive relative density contains a difference which is a square.

https://arxiv.org/abs/math/0610050

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