Found with PFGW Version 3.7.5

1001^6438-6438^1001 is 2-PRP! (11.3947s+0.0026s)
1001^6438-6438^1001 is 3-PRP! (11.7632s+0.0026s)
1001^6438-6438^1001 is 5-PRP! (12.5415s+0.0027s)
1001^6438-6438^1001 is 7-PRP! (14.5584s+0.0078s)
1001^6438-6438^1001 is 11-PRP! (12.1050s+0.0022s)

Primality testing 1001^6438-6438^1001 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 11
Running N-1 test using base 43
Running N+1 test using discriminant 53, base 3+sqrt(53)
Calling N-1 BLS with factored part 0.07% and helper 0.02% (0.24% proof)
1001^6438-6438^1001 is Fermat and Lucas PRP! (82.7967s+0.0034s)

To be completed...