Does anyone know how to prove or disprove that for all even integers $k$ that there exists a prime number $n$ such that $n+k$ & $n-k$ are both prime?

Question

This problem came into my head while working on something similar. There was a similar question asked, but it was something slightly different. I’ve been stuck on this,unable to get any progress since I’m on holiday. I first thought about the fact that there are infinitely many primes, and that if these were randomly dispersed that the problem above should be true, but realised that that hasn’t been solved, then I moved onto brute force, but found no counterexample, asked ai and discord server and still no progress. Anyone got anything at all?

Answer 1

Your statement is false for $k = 16$. 3 must divide one of the numbers $n - 16$, $n$, $n + 16$. This is only possible if $n - 16 = 3$, $n = 19$, $n + 16 = 35$.