Found with PFGW Version 3.6.0

5163^6452-6452^5163 is 2-PRP! (23.3075s+0.0029s)
5163^6452-6452^5163 is 3-PRP! (23.7399s+0.0041s)
5163^6452-6452^5163 is 5-PRP! (23.8507s+0.0027s)
5163^6452-6452^5163 is 7-PRP! (23.4420s+0.0026s)
5163^6452-6452^5163 is 11-PRP! (23.3609s+0.0026s)

Primality testing 5163^6452-6452^5163 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 7
Running N+1 test using discriminant 19, base 2+sqrt(19)
Calling N+1 BLS with factored part 0.06% and helper 0.02% (0.20% proof)
5163^6452-6452^5163 is Fermat and Lucas PRP! (138.1346s+0.0042s)

To be completed...