Found with PFGW Version 3.7.5

5414^8305-8305^5414 is 2-PRP! (19.8993s+0.0016s)
5414^8305-8305^5414 is 3-PRP! (18.2491s+0.0016s)
5414^8305-8305^5414 is 5-PRP! (18.5072s+0.0018s)
5414^8305-8305^5414 is 7-PRP! (18.7800s+0.0016s)
5414^8305-8305^5414 is 11-PRP! (18.2046s+0.0014s)

Primality testing 5414^8305-8305^5414 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 6+sqrt(7)
Calling N-1 BLS with factored part 0.06% and helper 0.02% (0.21% proof)
5414^8305-8305^5414 is Fermat and Lucas PRP! (113.1523s+0.0024s)

To be completed...