Found with PFGW Version 3.7.5

999^9082-9082^999 is 2-PRP! (13.9212s+0.0012s)
999^9082-9082^999 is 3-PRP! (13.6751s+0.0012s)
999^9082-9082^999 is 5-PRP! (13.6569s+0.0013s)
999^9082-9082^999 is 7-PRP! (13.6569s+0.0013s)
999^9082-9082^999 is 11-PRP! (13.7807s+0.0013s)

Primality testing 999^9082-9082^999 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N+1 BLS with factored part 0.10% and helper 0.04% (0.35% proof)
999^9082-9082^999 is Fermat and Lucas PRP! (72.1122s+0.0020s)

To be completed...