Found with PFGW Version 3.7.5

6374^9995-9995^6374 is 2-PRP! (26.3774s+0.0016s)
6374^9995-9995^6374 is 3-PRP! (26.5266s+0.0021s)
6374^9995-9995^6374 is 5-PRP! (26.2556s+0.0019s)
6374^9995-9995^6374 is 7-PRP! (26.6239s+0.0016s)
6374^9995-9995^6374 is 11-PRP! (26.5788s+0.0017s)

Primality testing 6374^9995-9995^6374 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 2+sqrt(13)
Calling N-1 BLS with factored part 0.04% and helper 0.02% (0.14% proof)
6374^9995-9995^6374 is Fermat and Lucas PRP! (178.4621s+0.0026s)

To be completed...