Found with PFGW Version 3.8.0

7178^11011-11011^7178 is 2-PRP! (34.8278s+0.0069s)
7178^11011-11011^7178 is 3-PRP! (34.8119s+0.0076s)
7178^11011-11011^7178 is 5-PRP! (34.8758s+0.0065s)
7178^11011-11011^7178 is 7-PRP! (34.7194s+0.0075s)
7178^11011-11011^7178 is 11-PRP! (34.7721s+0.0065s)

Primality testing 7178^11011-11011^7178 [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 0.02% and helper 0.00% (0.05% proof)
7178^11011-11011^7178 is Fermat and Lucas PRP! (175.8440s+0.0075s)

To be completed...