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? @AndréNicolas, My question is if there are infinitely many primes with difference $2$, Is there a solid relation between that conjecture and the conjecture that there are infinitely many with difference $4,6,8,10,...$? And the same if there are finite. I'm having trouble wording the question and would really appreciate it if you edited the question if you understood me, thanks.