TITLE: Prime plus square equals prime
QUESTION [8 upvotes]: 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

REPLY [4 votes]: As Joel mentions, this follows from the work of Tao and Ziegler.
Alternatively, this can be directly deduced from the density of the primes and the known bounds on the Furstenberg–Sárközy theorem. Indeed, the bounds of Pintz, Steiger, and Szemeredi from the 1980's are suffient. See:
J. Pintz, W. L. Steiger, and E. Szemeredi, On sets of natural numbers whose difference set contains no squares, J. London Math. Soc. (2) 37 (1988), 219–231
See also the even stronger bounds in the more recent work of Bloom and Maynard: https://arxiv.org/pdf/2011.13266.pdf.