How many pairs of prime number $(p,q)$ so that $p-q$ is a square number?

Question

How many pairs of prime number $(p,q)$ so that $p-q$ is a square number ? Moreover, how many pairs of prime number $(p,q)$ so that $p-q$ is a $k$th power of a number or a power of $n$ with $k,n \geq 2$ ?

Answer 1

It is conjectured that for every even positive integer $d$ there are infinitely many pairs of primes $p,q$ with $p-q=d$. However, this is still open, as are your questions.

See OEIS sequence A065376 for primes of the form $p+k^2$ and OEIS sequence A065380 for primes of the form $p+2^k$.