Found with PFGW Version 3.6.0

1355^3864-3864^1355 is 2-PRP! (5.4970s+0.0033s)
1355^3864-3864^1355 is 3-PRP! (5.4800s+0.0032s)
1355^3864-3864^1355 is 5-PRP! (5.4700s+0.0034s)
1355^3864-3864^1355 is 7-PRP! (5.4762s+0.0018s)
1355^3864-3864^1355 is 11-PRP! (5.4836s+0.0017s)

Primality testing 1355^3864-3864^1355 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N-1 test using base 17
Running N+1 test using discriminant 37, base 12+sqrt(37)
Calling N-1 BLS with factored part 0.04% and helper 0.00% (0.14% proof)
1355^3864-3864^1355 is Fermat and Lucas PRP! (33.5784s+0.0027s)

To be completed...