Found with PFGW Version 3.8.0

588^13231-13231^588 is 2-PRP! (24.8125s+0.0046s)
588^13231-13231^588 is 3-PRP! (24.1077s+0.0045s)
588^13231-13231^588 is 5-PRP! (23.9677s+0.0041s)
588^13231-13231^588 is 7-PRP! (23.9808s+0.0043s)
588^13231-13231^588 is 11-PRP! (23.9318s+0.0119s)

Primality testing 588^13231-13231^588 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N-1 test using base 19
Running N+1 test using discriminant 71, base 1+sqrt(71)
Calling N-1 BLS with factored part 0.04% and helper 0.03% (0.14% proof)
588^13231-13231^588 is Fermat and Lucas PRP! (149.4439s+0.0067s)

To be completed...