Found with PFGW Version 3.8.0

10306^12885-12885^10306 is 2-PRP! (48.2533s+0.0030s)
10306^12885-12885^10306 is 3-PRP! (49.6603s+0.0018s)
10306^12885-12885^10306 is 5-PRP! (48.2430s+0.0020s)
10306^12885-12885^10306 is 7-PRP! (51.1797s+0.0026s)
10306^12885-12885^10306 is 11-PRP! (49.1953s+0.0021s)

Primality testing 10306^12885-12885^10306 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 13, base 2+sqrt(13)
Calling N-1 BLS with factored part 0.03% and helper 0.02% (0.10% proof)
10306^12885-12885^10306 is Fermat and Lucas PRP! (282.2747s+0.0069s)

To be completed...