Found with PFGW Version 3.8.0

5146^12777-12777^5146 is 2-PRP! (42.3879s+0.0070s)
5146^12777-12777^5146 is 3-PRP! (43.6339s+0.0079s)
5146^12777-12777^5146 is 5-PRP! (42.2846s+0.0074s)
5146^12777-12777^5146 is 7-PRP! (42.4213s+0.0085s)
5146^12777-12777^5146 is 11-PRP! (42.1904s+0.0147s)

Primality testing 5146^12777-12777^5146 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 13, base 2+sqrt(13)
Calling N+1 BLS with factored part 0.02% and helper 0.00% (0.06% proof)
5146^12777-12777^5146 is Fermat and Lucas PRP! (240.6355s+0.0087s)

To be completed...