Found with PFGW Version 3.7.5

5390^8889-8889^5390 is 2-PRP! (21.4645s+0.0014s)
5390^8889-8889^5390 is 3-PRP! (21.4618s+0.0014s)
5390^8889-8889^5390 is 5-PRP! (21.5858s+0.0016s)
5390^8889-8889^5390 is 7-PRP! (21.5928s+0.0016s)
5390^8889-8889^5390 is 11-PRP! (21.7070s+0.0026s)

Primality testing 5390^8889-8889^5390 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Running N+1 test using discriminant 29, base 1+sqrt(29)
Calling N+1 BLS with factored part 0.04% and helper 0.01% (0.14% proof)
5390^8889-8889^5390 is Fermat and Lucas PRP! (106.8757s+0.0023s)

To be completed...