The following question is equivalent to a problem in group theory. Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid a$ or $r^2\mid a$ and specially $ r^3 $ does not divide $a$ and also $(1+kr)\not\mid a$, for each nonzero $k$?
We check it for many primes as the computer allows us. Always we get the positive answer. Thanks for your answers.