Found with PFGW Version 3.7.5

8384^8423-8423^8384 is 2-PRP! (20.8066s+0.0013s)
8384^8423-8423^8384 is 3-PRP! (21.4345s+0.0010s)
8384^8423-8423^8384 is 5-PRP! (21.5596s+0.0012s)
8384^8423-8423^8384 is 7-PRP! (21.4469s+0.0011s)
8384^8423-8423^8384 is 11-PRP! (21.2547s+0.0008s)

Primality testing 8384^8423-8423^8384 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 2+sqrt(11)
Calling N+1 BLS with factored part 0.03% and helper 0.01% (0.09% proof)
8384^8423-8423^8384 is Fermat and Lucas PRP! (106.4228s+0.0022s)

To be completed...