Found with PFGW Version 3.8.0

5436^12865-12865^5436 is 2-PRP! (43.1589s+0.0067s)
5436^12865-12865^5436 is 3-PRP! (42.8980s+0.0064s)
5436^12865-12865^5436 is 5-PRP! (42.8942s+0.0052s)
5436^12865-12865^5436 is 7-PRP! (42.7338s+0.0051s)
5436^12865-12865^5436 is 11-PRP! (42.7078s+0.0056s)

Primality testing 5436^12865-12865^5436 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N+1 test using discriminant 17, base 1+sqrt(17)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.06% proof)
5436^12865-12865^5436 is Fermat and Lucas PRP! (244.5714s+0.0087s)

To be completed...