Found with PFGW Version 3.6.0

1005^3634-3634^1005 is 2-PRP! (5.4895s+0.0017s)
1005^3634-3634^1005 is 3-PRP! (5.4859s+0.0018s)
1005^3634-3634^1005 is 5-PRP! (5.5102s+0.0027s)
1005^3634-3634^1005 is 7-PRP! (5.5173s+0.0018s)
1005^3634-3634^1005 is 11-PRP! (5.4816s+0.0031s)

Primality testing 1005^3634-3634^1005 [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 0.09% and helper 0.05% (0.31% proof)
1005^3634-3634^1005 is Fermat and Lucas PRP! (24.9719s+0.0045s)

To be completed...