Found with PFGW Version 3.7.5

3413^8172-8172^3413 is 2-PRP! (34.4950s+0.0044s)
3413^8172-8172^3413 is 3-PRP! (34.5859s+0.0049s)
3413^8172-8172^3413 is 5-PRP! (33.5902s+0.0036s)
3413^8172-8172^3413 is 7-PRP! (33.9106s+0.0039s)
3413^8172-8172^3413 is 11-PRP! (34.6205s+0.0041s)

Primality testing 3413^8172-8172^3413 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 6+sqrt(13)
Calling N-1 BLS with factored part 0.04% and helper 0.04% (0.16% proof)
3413^8172-8172^3413 is Fermat and Lucas PRP! (172.2195s+0.0049s)

To be completed...