Found with PFGW Version 3.7.5

341^6564-6564^341 is 2-PRP! (9.4626s+0.0016s)
341^6564-6564^341 is 3-PRP! (10.6427s+0.0026s)
341^6564-6564^341 is 5-PRP! (12.5058s+0.0017s)
341^6564-6564^341 is 7-PRP! (11.5818s+0.0019s)
341^6564-6564^341 is 11-PRP! (11.7206s+0.0017s)

Primality testing 341^6564-6564^341 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 8+sqrt(13)
Calling N+1 BLS with factored part 0.03% and helper 0.01% (0.10% proof)
341^6564-6564^341 is Fermat and Lucas PRP! (53.5913s+0.0040s)

To be completed...