Found with PFGW Version 3.7.5

5589^10238-10238^5589 is 2-PRP! (26.4502s+0.0017s)
5589^10238-10238^5589 is 3-PRP! (26.5998s+0.0015s)
5589^10238-10238^5589 is 5-PRP! (26.8775s+0.0015s)
5589^10238-10238^5589 is 7-PRP! (26.5145s+0.0016s)
5589^10238-10238^5589 is 11-PRP! (26.9802s+0.0017s)

Primality testing 5589^10238-10238^5589 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 6+sqrt(13)
Calling N+1 BLS with factored part 0.02% and helper 0.01% (0.07% proof)
5589^10238-10238^5589 is Fermat and Lucas PRP! (162.8444s+0.0025s)

To be completed...