Found with PFGW Version 3.6.0

4065^4556-4556^4065 is 2-PRP! (11.8814s+0.0035s)
4065^4556-4556^4065 is 3-PRP! (11.8067s+0.0038s)
4065^4556-4556^4065 is 5-PRP! (11.7920s+0.0039s)
4065^4556-4556^4065 is 7-PRP! (11.7856s+0.0031s)
4065^4556-4556^4065 is 11-PRP! (12.0514s+0.0024s)

Primality testing 4065^4556-4556^4065 [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 discriminant 19, base 14+sqrt(19)
Calling N-1 BLS with factored part 0.12% and helper 0.02% (0.39% proof)
4065^4556-4556^4065 is Fermat and Lucas PRP! (79.7734s+0.0021s)

To be completed...