Found with PFGW Version 3.8.0

44^29191-29191^44 is 2-PRP! (43.4080s+0.0088s)
44^29191-29191^44 is 3-PRP! (43.1350s+0.0088s)
44^29191-29191^44 is 5-PRP! (43.2236s+0.0092s)
44^29191-29191^44 is 7-PRP! (43.1438s+0.0089s)
44^29191-29191^44 is 11-PRP! (43.2121s+0.0090s)

Primality testing 44^29191-29191^44 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 2+sqrt(13)
Calling N+1 BLS with factored part 0.03% and helper 0.02% (0.11% proof)
44^29191-29191^44 is Fermat and Lucas PRP! (290.5202s+0.0098s)

To be completed...