Found with PFGW Version 3.6.0

5166^6295-6295^5166 is 2-PRP! (22.8430s+0.0027s)
5166^6295-6295^5166 is 3-PRP! (23.1129s+0.0036s)
5166^6295-6295^5166 is 5-PRP! (23.0898s+0.0026s)
5166^6295-6295^5166 is 7-PRP! (23.0924s+0.0038s)
5166^6295-6295^5166 is 11-PRP! (23.1138s+0.0040s)

Primality testing 5166^6295-6295^5166 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N+1 BLS with factored part 0.03% and helper 0.02% (0.12% proof)
5166^6295-6295^5166 is Fermat and Lucas PRP! (111.4142s+0.0025s)

To be completed...