Found with PFGW Version 3.7.5

11376^12181+12181^11376 is 2-PRP! (156.2719s+0.0084s)
11376^12181+12181^11376 is 3-PRP! (158.1105s+0.0097s)
11376^12181+12181^11376 is 5-PRP! (159.1182s+0.0094s)
11376^12181+12181^11376 is 7-PRP! (158.5592s+0.0075s)
11376^12181+12181^11376 is 11-PRP! (152.5867s+0.0085s)

Primality testing 11376^12181+12181^11376 [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 17, base 6+sqrt(17)
Calling N-1 BLS with factored part 0.02% and helper 0.00% (0.07% proof)
11376^12181+12181^11376 is Fermat and Lucas PRP! (1128.6589s+0.0077s)

To be completed...