Found with PFGW Version 3.6.0

6782^6851-6851^6782 is 2-PRP! (28.8505s+0.0043s)
6782^6851-6851^6782 is 3-PRP! (28.7859s+0.0042s)
6782^6851-6851^6782 is 5-PRP! (29.0105s+0.0047s)
6782^6851-6851^6782 is 7-PRP! (28.9930s+0.0029s)
6782^6851-6851^6782 is 11-PRP! (28.8381s+0.0032s)

Primality testing 6782^6851-6851^6782 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 10+sqrt(7)
Calling N+1 BLS with factored part 0.05% and helper 0.00% (0.15% proof)
6782^6851-6851^6782 is Fermat and Lucas PRP! (176.1301s+0.0077s)

To be completed...