Found with PFGW Version 3.7.5

5023^7004-7004^5023 is 2-PRP! (20.8046s+0.0027s)
5023^7004-7004^5023 is 3-PRP! (21.0771s+0.0026s)
5023^7004-7004^5023 is 5-PRP! (21.5535s+0.0031s)
5023^7004-7004^5023 is 7-PRP! (21.2617s+0.0027s)
5023^7004-7004^5023 is 11-PRP! (21.2228s+0.0022s)

Primality testing 5023^7004-7004^5023 [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 0.03% (0.26% proof)
5023^7004-7004^5023 is Fermat and Lucas PRP! (122.1572s+0.0043s)

To be completed...