Found with PFGW Version 3.7.5

314^6291-6291^314 is 2-PRP! (10.3947s+0.0008s)
314^6291-6291^314 is 3-PRP! (8.4508s+0.0006s)
314^6291-6291^314 is 5-PRP! (8.6279s+0.0004s)
314^6291-6291^314 is 7-PRP! (8.9605s+0.0004s)
314^6291-6291^314 is 11-PRP! (8.4979s+0.0007s)

Primality testing 314^6291-6291^314 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 3+sqrt(11)
Calling N+1 BLS with factored part 0.07% and helper 0.06% (0.25% proof)
314^6291-6291^314 is Fermat and Lucas PRP! (38.9654s+0.0109s)

To be completed...