Found with PFGW Version 3.8.0

4004^13239-13239^4004 is 2-PRP! (42.7594s+0.0011s)
4004^13239-13239^4004 is 3-PRP! (42.3674s+0.0010s)
4004^13239-13239^4004 is 5-PRP! (42.5830s+0.0010s)
4004^13239-13239^4004 is 7-PRP! (42.4236s+0.0013s)
4004^13239-13239^4004 is 11-PRP! (42.3519s+0.0010s)

Primality testing 4004^13239-13239^4004 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 31, base 2+sqrt(31)
Calling N+1 BLS with factored part 0.04% and helper 0.02% (0.14% proof)
4004^13239-13239^4004 is Fermat and Lucas PRP! (243.2515s+0.0132s)

To be completed...