Primes p such that p^2+p+1=q is prime, and p is largest prime factor of q^31
I was wondering if there's even such a prime p that fits the following:
1. p^2+p+1=q is prime (seems that p must be of the form 6n1, other than 2 and 3).
2. Largest prime factor of q^31 is p.
So far, I haven't found any that fit #2 (searched up to p=997), but I'm guessing #1 has infinitely many solutions. Anyone know how I might do this in PARI/gp? I want to search up to at least p=9999991.
