Found with PFGW Version 3.7.5

4258^8643-8643^4258 is 2-PRP! (17.6498s+0.0013s)
4258^8643-8643^4258 is 3-PRP! (17.6296s+0.0013s)
4258^8643-8643^4258 is 5-PRP! (17.6843s+0.0014s)
4258^8643-8643^4258 is 7-PRP! (17.5698s+0.0013s)
4258^8643-8643^4258 is 11-PRP! (17.6588s+0.0015s)

Primality testing 4258^8643-8643^4258 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 2+sqrt(13)
Calling N-1 BLS with factored part 0.06% and helper 0.00% (0.18% proof)
4258^8643-8643^4258 is Fermat and Lucas PRP! (118.8184s+0.0022s)

To be completed...