Found with PFGW Version 3.6.0

4440^5053-5053^4440 is 2-PRP! (13.7694s+0.0020s)
4440^5053-5053^4440 is 3-PRP! (14.6424s+0.0020s)
4440^5053-5053^4440 is 5-PRP! (14.4201s+0.0021s)
4440^5053-5053^4440 is 7-PRP! (14.2789s+0.0019s)
4440^5053-5053^4440 is 11-PRP! (14.6216s+0.0020s)

Primality testing 4440^5053-5053^4440 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N-1 BLS with factored part 0.05% and helper 0.05% (0.19% proof)
4440^5053-5053^4440 is Fermat and Lucas PRP! (69.5298s+0.0050s)

To be completed...