Found with PFGW Version 3.6.0

6569^10122-10122^6569 is 2-PRP! (65.1467s+0.0032s)
6569^10122-10122^6569 is 3-PRP! (65.2002s+0.0032s)
6569^10122-10122^6569 is 5-PRP! (65.2537s+0.0033s)
6569^10122-10122^6569 is 7-PRP! (65.2482s+0.0032s)
6569^10122-10122^6569 is 11-PRP! (65.1492s+0.0032s)

Primality testing 6569^10122-10122^6569 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N+1 test using discriminant 29, base 4+sqrt(29)
Calling N-1 BLS with factored part 0.02% and helper 0.02% (0.08% proof)
6569^10122-10122^6569 is Fermat and Lucas PRP! (385.8162s+0.0036s)

To be completed...