Found with PFGW Version 3.7.5

39^40359-38 is 2-PRP! (36.6991s+0.0036s)
39^40359-38 is 3-PRP! (36.5943s+0.0039s)
39^40359-38 is 5-PRP! (38.6456s+0.0060s)
39^40359-38 is 7-PRP! (42.3323s+0.0054s)
39^40359-38 is 11-PRP! (37.3606s+0.0036s)

Primality testing 39^40359-38 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N-1 test using base 11
Running N-1 test using base 17
Running N+1 test using discriminant 43, base 1+sqrt(43)
Calling N-1 BLS with factored part 0.04% and helper 0.02% (0.13% proof)
39^40359-38 is Fermat and Lucas PRP! (371.7023s+0.0055s)

To be completed...