Found with PFGW Version 3.7.5

6351^7904-7904^6351 is 2-PRP! (36.3298s+0.0034s)
6351^7904-7904^6351 is 3-PRP! (35.2005s+0.0047s)
6351^7904-7904^6351 is 5-PRP! (36.6175s+0.0040s)
6351^7904-7904^6351 is 7-PRP! (36.5462s+0.0047s)
6351^7904-7904^6351 is 11-PRP! (36.2249s+0.0042s)

Primality testing 6351^7904-7904^6351 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 7
Running N+1 test using discriminant 37, base 2+sqrt(37)
Calling N-1 BLS with factored part 0.07% and helper 0.03% (0.24% proof)
6351^7904-7904^6351 is Fermat and Lucas PRP! (226.3167s+0.0057s)

To be completed...