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
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.