Found with PFGW Version 3.5.2

2538^3163-3163^2538 is 2-PRP! (21.8231s+0.0057s)
2538^3163-3163^2538 is 3-PRP! (25.7944s+0.0077s)
2538^3163-3163^2538 is 5-PRP! (25.4490s+0.0064s)
2538^3163-3163^2538 is 7-PRP! (24.9714s+0.0069s)
2538^3163-3163^2538 is 11-PRP! (25.1991s+0.0065s)

Primality testing 2538^3163-3163^2538 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N+1 BLS with factored part 0.04% and helper 0.00% (0.14% proof)
2538^3163-3163^2538 is Fermat and Lucas PRP! (127.4659s+0.0258s)

To be completed...