Found with PFGW Version 3.4.8

44^9679-43 is 2-PRP! (15.6482s+0.0068s)
44^9679-43 is 3-PRP! (16.4267s+0.0082s)
44^9679-43 is 5-PRP! (15.2865s+0.0048s)
44^9679-43 is 7-PRP! (15.9960s+0.0073s)
44^9679-43 is 11-PRP! (15.7541s+0.0054s)

Primality testing 44^9679-43 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 17, base 3+sqrt(17)
Calling N-1 BLS with factored part 0.15% and helper 0.01% (0.47% proof)
44^9679-43 is Fermat and Lucas PRP! (99.9193s+0.0130s)

To be completed...