r(n)= n-th repunit and p(n)=n-th prime.

PFGW Version 1.2.0 for Windows [FFT v23.8]

Primality testing 3414!*r(3416)+p(3421) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 17, base 1+sqrt(17)
3414!*r(3416)+p(3421) is Fermat and Lucas PRP!

To be completed...