Found with PFGW Version 3.7.5

2191^5582-5582^2191 is 2-PRP! (11.9040s+0.0017s)
2191^5582-5582^2191 is 3-PRP! (13.7482s+0.0021s)
2191^5582-5582^2191 is 5-PRP! (13.8311s+0.0016s)
2191^5582-5582^2191 is 7-PRP! (13.3725s+0.0017s)
2191^5582-5582^2191 is 11-PRP! (13.8908s+0.0018s)

Primality testing 2191^5582-5582^2191 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling N-1 BLS with factored part 0.03% and helper 0.02% (0.10% proof)
2191^5582-5582^2191 is Fermat and Lucas PRP! (65.7374s+0.0031s)

To be completed...