Found with PFGW Version 3.7.5

4407^8002-8002^4407 is 2-PRP! (24.3271s+0.0021s)
4407^8002-8002^4407 is 3-PRP! (24.7857s+0.0021s)
4407^8002-8002^4407 is 5-PRP! (24.1546s+0.0021s)
4407^8002-8002^4407 is 7-PRP! (24.3256s+0.0022s)
4407^8002-8002^4407 is 11-PRP! (24.8585s+0.0023s)

Primality testing 4407^8002-8002^4407 [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 0.02% and helper 0.01% (0.06% proof)
4407^8002-8002^4407 is Fermat and Lucas PRP! (126.5275s+0.0037s)

To be completed...