Found with PFGW Version 3.7.5

365^5276-5276^365 is 2-PRP! (7.3714s+0.0132s)
365^5276-5276^365 is 3-PRP! (7.5889s+0.0038s)
365^5276-5276^365 is 5-PRP! (6.9344s+0.0020s)
365^5276-5276^365 is 7-PRP! (7.4013s+0.0021s)
365^5276-5276^365 is 11-PRP! (8.0009s+0.0018s)

Primality testing 365^5276-5276^365 [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 0.08% and helper 0.07% (0.33% proof)
365^5276-5276^365 is Fermat and Lucas PRP! (36.8889s+0.0034s)

To be completed...