Found with PFGW Version 3.8.0

12950^13051-13051^12950 is 2-PRP! (55.0933s+0.0097s)
12950^13051-13051^12950 is 3-PRP! (56.3975s+0.0067s)
12950^13051-13051^12950 is 5-PRP! (55.0789s+0.0095s)
12950^13051-13051^12950 is 7-PRP! (56.2278s+0.0093s)
12950^13051-13051^12950 is 11-PRP! (55.0845s+0.0104s)

Primality testing 12950^13051-13051^12950 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 23
Running N+1 test using discriminant 41, base 1+sqrt(41)
Calling N-1 BLS with factored part 0.03% and helper 0.02% (0.10% proof)
12950^13051-13051^12950 is Fermat and Lucas PRP! (331.2636s+0.0093s)

To be completed...