Found with PFGW Version 3.7.5

7174^8349-8349^7174 is 2-PRP! (19.3262s+0.0017s)
7174^8349-8349^7174 is 3-PRP! (18.9218s+0.0018s)
7174^8349-8349^7174 is 5-PRP! (18.8548s+0.0019s)
7174^8349-8349^7174 is 7-PRP! (18.9689s+0.0018s)
7174^8349-8349^7174 is 11-PRP! (18.9177s+0.0017s)

Primality testing 7174^8349-8349^7174 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 6+sqrt(7)
Calling N+1 BLS with factored part 0.01% and helper 0.00% (0.04% proof)
7174^8349-8349^7174 is Fermat and Lucas PRP! (117.4743s+0.0026s)

To be completed...