TITLE: If $p_1$ and $p_2$ are primes, prove that there exists an integer such that $p_1+k$ is a prime but $p_2+k$ is not.
QUESTION [11 upvotes]: Let $P_{1}$ and $P_{2}$ be primes. Prove that there exists a $k\in \Bbb Z^+$ such that $P_{1} + k$ is a prime but $P_{2} + k$ is not.

It seems like the proof should be trivial, but for some reason, I am not able to construct it. 
Any help would be appreciated.

REPLY [3 votes]: Assume wlog $p_1 < p_2$ and $p_1+k$ is prime iff $p_2+k$ is prime. Set $z:=p_2-p_1$. By induction, $p_1+nz$ is prime for all $n$. This is impossible since the primes have density zero.