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. Oct 2, 2018 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). Oct 2, 2018 at 17:45
• I think, should such a result be established, that this square can be $4$. Googling "jumping champions" may be insightful. Oct 2, 2018 at 18:08