Found with PFGW Version 3.6.0

1083^4418-4418^1083 is 2-PRP! (8.9757s+0.0032s)
1083^4418-4418^1083 is 3-PRP! (8.9887s+0.0017s)
1083^4418-4418^1083 is 5-PRP! (8.9771s+0.0033s)
1083^4418-4418^1083 is 7-PRP! (8.9859s+0.0020s)
1083^4418-4418^1083 is 11-PRP! (9.0022s+0.0019s)

Primality testing 1083^4418-4418^1083 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 10+sqrt(13)
Calling N+1 BLS with factored part 0.09% and helper 0.07% (0.33% proof)
1083^4418-4418^1083 is Fermat and Lucas PRP! (51.1679s+0.0018s)

To be completed...