Q:

For every pair of distinct primes $p_1$ and $p_2$, there are $z$ such that $p_1 + z$ is prime but $p_2 + z$ is not

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.

A:

Apply Dirichlet's Theorem on prime numbers in arithmetic progression. For $z$ try all possible multiples of $p_1$. The $z+p_1$ will always be composite; but $p_2+xp_1$, for various positive integers $x$ represents an arithmetic progression of common difference $p_1$, with $p_2$ as first term as $\gcd(p_1,p_2)=1$ we are done.

A:

Let $r=p_2-p_1$. There are infinitely many primes $p$ such that $p+1,p+2,\ldots,p+r$ are composite: take for example the greatest prime lesser than $s!+2$, for each $s>r$.

