Prime plus square equals prime

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

• Almost assuredly, and that square might be less than 246. Gerhard "See About Small Prime Gaps" Paseman, 2018.10.02. – Gerhard Paseman Oct 2 '18 at 17:01
• 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). – Stoyan Apostolov Oct 2 '18 at 17:45
• I think, should such a result be established, that this square can be $4$. Googling "jumping champions" may be insightful. – Sylvain JULIEN Oct 2 '18 at 18:08

1 Answer

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