Found with PFGW Version 3.7.5

11508^12391+12391^11508 is 2-PRP! (135.3517s+0.0051s)
11508^12391+12391^11508 is 3-PRP! (132.4153s+0.0058s)
11508^12391+12391^11508 is 5-PRP! (132.8158s+0.0050s)
11508^12391+12391^11508 is 7-PRP! (117.6554s+0.0049s)
11508^12391+12391^11508 is 11-PRP! (124.4006s+0.0058s)

Primality testing 11508^12391+12391^11508 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 11
Running N+1 test using discriminant 31, base 1+sqrt(31)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.08% proof)
11508^12391+12391^11508 is Fermat and Lucas PRP! (877.4444s+0.0061s)

To be completed...