Found with PFGW Version 3.8.0

314^13049-13049^314 is 2-PRP! (20.0378s+0.0007s)
314^13049-13049^314 is 3-PRP! (18.1857s+0.0006s)
314^13049-13049^314 is 5-PRP! (18.2153s+0.0006s)
314^13049-13049^314 is 7-PRP! (18.2502s+0.0006s)
314^13049-13049^314 is 11-PRP! (18.3522s+0.0005s)

Primality testing 314^13049-13049^314 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 2+sqrt(11)
Calling N+1 BLS with factored part 0.01% and helper 0.00% (0.05% proof)
314^13049-13049^314 is Fermat and Lucas PRP! (106.4712s+0.0050s)

To be completed...