Found with PFGW Version 3.6.0

398^5023-5023^398 is 2-PRP! (6.1779s+0.0017s)
398^5023-5023^398 is 3-PRP! (6.1796s+0.0017s)
398^5023-5023^398 is 5-PRP! (6.1808s+0.0016s)
398^5023-5023^398 is 7-PRP! (6.1794s+0.0015s)
398^5023-5023^398 is 11-PRP! (6.1909s+0.0018s)

Primality testing 398^5023-5023^398 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 6+sqrt(7)
Calling N+1 BLS with factored part 0.08% and helper 0.03% (0.28% proof)
398^5023-5023^398 is Fermat and Lucas PRP! (36.7710s+0.0016s)

To be completed...