Found with PFGW Version 3.7.5

2781^8588-8588^2781 is 2-PRP! (15.7202s+0.0013s)
2781^8588-8588^2781 is 3-PRP! (15.5538s+0.0012s)
2781^8588-8588^2781 is 5-PRP! (16.2451s+0.0019s)
2781^8588-8588^2781 is 7-PRP! (15.5528s+0.0014s)
2781^8588-8588^2781 is 11-PRP! (15.5940s+0.0013s)

Primality testing 2781^8588-8588^2781 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling N+1 BLS with factored part 0.04% and helper 0.03% (0.14% proof)
2781^8588-8588^2781 is Fermat and Lucas PRP! (77.8582s+0.0025s)

To be completed...