Found with PFGW Version 3.8.0

500^13689-13689^500 is 2-PRP! (24.2983s+0.0042s)
500^13689-13689^500 is 3-PRP! (28.0544s+0.0269s)
500^13689-13689^500 is 5-PRP! (26.1686s+0.0040s)
500^13689-13689^500 is 7-PRP! (26.6160s+0.0038s)
500^13689-13689^500 is 11-PRP! (24.2077s+0.0045s)

Primality testing 500^13689-13689^500 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N+1 BLS with factored part 0.03% and helper 0.00% (0.10% proof)
500^13689-13689^500 is Fermat and Lucas PRP! (120.6375s+0.0072s)

To be completed...