Found with PFGW Version 3.7.5

8482^12591+12591^8482 is 2-PRP! (47.2591s+0.0022s)
8482^12591+12591^8482 is 3-PRP! (46.9574s+0.0022s)
8482^12591+12591^8482 is 5-PRP! (46.7764s+0.0022s)
8482^12591+12591^8482 is 7-PRP! (55.2686s+0.0027s)
8482^12591+12591^8482 is 11-PRP! (52.8185s+0.0024s)

Primality testing 8482^12591+12591^8482 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N-1 test using base 19
Running N+1 test using discriminant 29, base 11+sqrt(29)
Calling N+1 BLS with factored part 0.02% and helper 0.01% (0.09% proof)
8482^12591+12591^8482 is Fermat and Lucas PRP! (330.5804s+0.0022s)

To be completed...