Found with PFGW Version 3.8.0

771^25718+25718^771 is 2-PRP! (105.5908s+0.0055s)
771^25718+25718^771 is 3-PRP! (105.8433s+0.0046s)
771^25718+25718^771 is 5-PRP! (106.7654s+0.0047s)
771^25718+25718^771 is 7-PRP! (105.6985s+0.0047s)
771^25718+25718^771 is 11-PRP! (106.5081s+0.0050s)

Primality testing 771^25718+25718^771 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.06% proof)
771^25718+25718^771 is Fermat and Lucas PRP! (715.5298s+0.0060s)

To be completed...