If the set of primes where $p$, $p+2$ is infinite, would this imply that the set of $p$ and $p+2n$ is also infinite?

Question

If the set of primes $p$ such that $p+2$ is also prime is infinite, would this imply that the set of primes such that $p+2n$ where $n$ is any positive integer for each pair is also infinite?

Answer 1

This could be. It could be that a proof that there are infinitely many primes p and p+2 would imply the proof that there are infinetely many primes p and p+2n for all n = 1,2,3,4,... This is also called sometimes Polignac conjecture.

Answer 2

So far as I know, no one has ever proved anything along the lines of, "If there are infinitely many pairs of primes differing by $2$, then there are infinitely many pairs of primes differing by $4$."

On the other hand, I don't see what's so special about $2$ (in this context), and I bet that if the day comes when someone produces a proof for $2$, the techniques of that proof will also work for $2n$ generally.

Answer 3

I might be being stupid but surely no large enough twin prime pair (p,p+2) gives a prime pair of difference $4$.

So even if there were infinitely many twin primes, this would tell us nothing about the quantity of difference $4$ primes.