Found with PFGW Version 3.5.2

1004^3519-3519^1004 is 2-PRP! (22.7756s+0.0042s)
1004^3519-3519^1004 is 3-PRP! (24.6274s+0.0048s)
1004^3519-3519^1004 is 5-PRP! (20.5455s+0.0064s)
1004^3519-3519^1004 is 7-PRP! (24.7663s+0.0074s)
1004^3519-3519^1004 is 11-PRP! (24.4484s+0.0073s)

Primality testing 1004^3519-3519^1004 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 3+sqrt(11)
Calling N-1 BLS with factored part 0.14% and helper 0.11% (0.53% proof)
1004^3519-3519^1004 is Fermat and Lucas PRP! (133.6721s+0.0057s)

To be completed...