Found with PFGW Version 3.8.0

789^20158+20158^789 is 2-PRP! (64.3801s+0.0073s)
789^20158+20158^789 is 3-PRP! (64.2554s+0.0066s)
789^20158+20158^789 is 5-PRP! (64.3455s+0.0066s)
789^20158+20158^789 is 7-PRP! (64.4339s+0.0079s)
789^20158+20158^789 is 11-PRP! (64.2540s+0.0066s)

Primality testing 789^20158+20158^789 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N-1 test using base 19
Running N-1 test using base 37
Running N+1 test using discriminant 47, base 3+sqrt(47)
Calling N-1 BLS with factored part 0.01% and helper 0.01% (0.05% proof)
789^20158+20158^789 is Fermat and Lucas PRP! (466.1385s+0.0082s)

To be completed...