Found with PFGW Version 3.4.8

44^7638+43 is 2-PRP! (12.1187s+0.0209s)
44^7638+43 is 3-PRP! (12.6682s+0.0100s)
44^7638+43 is 5-PRP! (12.5370s+0.0120s)
44^7638+43 is 7-PRP! (12.2322s+0.0088s)
44^7638+43 is 11-PRP! (12.3178s+0.0183s)

Primality testing 44^7638+43 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 7, base 2+sqrt(7)
Calling N+1 BLS with factored part 0.06% and helper 0.06% (0.26% proof)
44^7638+43 is Fermat and Lucas PRP! (73.3361s+0.0254s)

To be completed...