Found with PFGW Version 3.7.5

3999^9544-9544^3999 is 2-PRP! (22.2581s+0.0013s)
3999^9544-9544^3999 is 3-PRP! (22.2322s+0.0014s)
3999^9544-9544^3999 is 5-PRP! (22.7101s+0.0015s)
3999^9544-9544^3999 is 7-PRP! (22.1325s+0.0013s)
3999^9544-9544^3999 is 11-PRP! (22.3494s+0.0014s)

Primality testing 3999^9544-9544^3999 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N-1 BLS with factored part 0.03% and helper 0.01% (0.11% proof)
3999^9544-9544^3999 is Fermat and Lucas PRP! (110.7134s+0.0023s)

To be completed...