Just a curiosity that occurred randomly. Is there a proof of the following:
For every two distinct primes $p_1$ and $p_2$, there exist an infinity of positive integers $z$ such that $p_1 + z$ is prime, but $p_2 + z$ is not prime.
It just "feels" true, but I don't know enough number theory to get an idea on how to prove this. I can't even guarantee that one such $z$ always exists.