Found with PFGW Version 3.7.5

257^29356+29356^257 is 2-PRP! (103.5197s+0.0042s)
257^29356+29356^257 is 3-PRP! (101.8723s+0.0027s)
257^29356+29356^257 is 5-PRP! (102.2546s+0.0028s)
257^29356+29356^257 is 7-PRP! (101.7592s+0.0028s)
257^29356+29356^257 is 11-PRP! (101.8233s+0.0026s)

Primality testing 257^29356+29356^257 [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 0.01% and helper 0.00% (0.03% proof)
257^29356+29356^257 is Fermat and Lucas PRP! (494.9812s+0.0026s)

To be completed...