Found with PFGW Version 3.7.5

5371^5640-5640^5371 is 2-PRP! (19.0022s+0.0020s)
5371^5640-5640^5371 is 3-PRP! (19.8155s+0.0028s)
5371^5640-5640^5371 is 5-PRP! (19.5156s+0.0022s)
5371^5640-5640^5371 is 7-PRP! (19.7986s+0.0020s)
5371^5640-5640^5371 is 11-PRP! (20.3141s+0.0021s)

Primality testing 5371^5640-5640^5371 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N-1 test using base 17
Running N-1 test using base 19
Running N+1 test using discriminant 31, base 10+sqrt(31)
Calling N+1 BLS with factored part 0.06% and helper 0.05% (0.25% proof)
5371^5640-5640^5371 is Fermat and Lucas PRP! (126.4120s+0.0034s)

To be completed...