Does there exist infinite primes $p$ such that either $(p^a-1)/2$ or $(p^a+1)/2$ is a prime power for some integer $a\geq 2$?



Find primes satisfying specific properties
For $(p^2+1)/2$ solutions are $7,41,239,63018038201,19175002942688032928599$.


For $(p^2-1)/2$ solution is $3$.


The Pell equation $x^2 - 2 y^2 = \pm 1$ has infinitely many integer solutions, not sure the primality constraints leaves only the above.


