Found with PFGW Version 3.7.5

11434^12977+12977^11434 is 2-PRP! (54.3979s+0.0027s)
11434^12977+12977^11434 is 3-PRP! (54.1588s+0.0029s)
11434^12977+12977^11434 is 5-PRP! (54.1543s+0.0026s)
11434^12977+12977^11434 is 7-PRP! (54.1854s+0.0026s)
11434^12977+12977^11434 is 11-PRP! (54.4376s+0.0028s)

Primality testing 11434^12977+12977^11434 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.08% proof)
11434^12977+12977^11434 is Fermat and Lucas PRP! (266.0674s+0.0025s)

To be completed...