Found with PFGW Version 3.7.5

5417^8288-8288^5417 is 2-PRP! (19.1614s+0.0008s)
5417^8288-8288^5417 is 3-PRP! (18.6193s+0.0008s)
5417^8288-8288^5417 is 5-PRP! (18.6345s+0.0011s)
5417^8288-8288^5417 is 7-PRP! (18.5516s+0.0009s)
5417^8288-8288^5417 is 11-PRP! (18.5636s+0.0009s)

Primality testing 5417^8288-8288^5417 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.07% proof)
5417^8288-8288^5417 is Fermat and Lucas PRP! (107.6713s+0.0110s)

To be completed...