Found with PFGW Version 3.7.5

5181^8258-8258^5181 is 2-PRP! (29.2710s+0.0052s)
5181^8258-8258^5181 is 3-PRP! (27.3687s+0.0023s)
5181^8258-8258^5181 is 5-PRP! (27.2334s+0.0024s)
5181^8258-8258^5181 is 7-PRP! (27.4495s+0.0023s)
5181^8258-8258^5181 is 11-PRP! (29.3442s+0.0037s)

Primality testing 5181^8258-8258^5181 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 17, base 8+sqrt(17)
Calling N+1 BLS with factored part 0.02% and helper 0.02% (0.08% proof)
5181^8258-8258^5181 is Fermat and Lucas PRP! (140.5606s+0.0055s)

To be completed...