Prime power conjecture
Tommy's prime power conjecture
Let p be a prime such that p−2 is also a prime.
Then there is always an integer n>1 such that
q_n = p^n−2
and
q_n is also a prime.
This is a special case of the generalized Bunyakovsky conjecture.
See also
and
https://math.stackexchange.com/questions/4841269/primes-of-the-form-q-n-pn-2
It is probably true for Eisenstein primes too.
https://en.wikipedia.org/wiki/Eisenstein_integer#Eisenstein_primes
picture : the Eisenstein primes.