For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Question

Yet another random conjecture about primes:

Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$.

Verified for all $p<100000$.

Answer 1

JasonM's observation is that the problem is equivalent to $q+1|p+1$. More generally, the fact that $p$ is prime is irrelevant because we just need to show there exists a $q$ for each $n$ such that $(q+1)|(4n+4)$. Trivially take $q=3$.