Found with PFGW Version 3.7.5

5089^5352-5352^5089 is 2-PRP! (14.9783s+0.0030s)
5089^5352-5352^5089 is 3-PRP! (16.6138s+0.0023s)
5089^5352-5352^5089 is 5-PRP! (15.7229s+0.0023s)
5089^5352-5352^5089 is 7-PRP! (15.7996s+0.0025s)
5089^5352-5352^5089 is 11-PRP! (16.6235s+0.0031s)

Primality testing 5089^5352-5352^5089 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 6+sqrt(13)
Calling N-1 BLS with factored part 0.11% and helper 0.03% (0.36% proof)
5089^5352-5352^5089 is Fermat and Lucas PRP! (80.7783s+0.0037s)

To be completed...