Found with PFGW Version 3.7.5

6229^8372-8372^6229 is 2-PRP! (19.0395s+0.0016s)
6229^8372-8372^6229 is 3-PRP! (18.7213s+0.0019s)
6229^8372-8372^6229 is 5-PRP! (18.5522s+0.0016s)
6229^8372-8372^6229 is 7-PRP! (18.5152s+0.0017s)
6229^8372-8372^6229 is 11-PRP! (18.6836s+0.0113s)

Primality testing 6229^8372-8372^6229 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 11+sqrt(11)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.07% proof)
6229^8372-8372^6229 is Fermat and Lucas PRP! (118.9241s+0.0025s)

To be completed...