Found with PFGW Version 3.7.5

38^17577-17577^38 is 2-PRP! (14.2195s+0.0017s)
38^17577-17577^38 is 3-PRP! (14.5168s+0.0020s)
38^17577-17577^38 is 5-PRP! (14.5521s+0.0021s)
38^17577-17577^38 is 7-PRP! (14.5447s+0.0021s)
38^17577-17577^38 is 11-PRP! (14.8425s+0.0020s)

Primality testing 38^17577-17577^38 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N+1 test using discriminant 31, base 1+sqrt(31)
Calling N+1 BLS with factored part 0.03% and helper 0.01% (0.12% proof)
38^17577-17577^38 is Fermat and Lucas PRP! (72.1469s+0.0019s)

To be completed...