Found with PFGW Version 3.7.5

6436^9165-9165^6436 is 2-PRP! (21.9548s+0.0016s)
6436^9165-9165^6436 is 3-PRP! (22.2846s+0.0017s)
6436^9165-9165^6436 is 5-PRP! (22.2263s+0.0016s)
6436^9165-9165^6436 is 7-PRP! (22.4296s+0.0017s)
6436^9165-9165^6436 is 11-PRP! (21.9986s+0.0015s)

Primality testing 6436^9165-9165^6436 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 9+sqrt(7)
Calling N-1 BLS with factored part 0.02% and helper 0.00% (0.08% proof)
6436^9165-9165^6436 is Fermat and Lucas PRP! (118.7251s+0.0026s)

To be completed...