Found with PFGW Version 3.7.5

5634^8899-8899^5634 is 2-PRP! (21.5928s+0.0015s)
5634^8899-8899^5634 is 3-PRP! (21.5233s+0.0016s)
5634^8899-8899^5634 is 5-PRP! (21.7259s+0.0015s)
5634^8899-8899^5634 is 7-PRP! (21.7238s+0.0014s)
5634^8899-8899^5634 is 11-PRP! (21.8206s+0.0016s)

Primality testing 5634^8899-8899^5634 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 17, base 1+sqrt(17)
Calling N+1 BLS with factored part 0.04% and helper 0.02% (0.13% proof)
5634^8899-8899^5634 is Fermat and Lucas PRP! (106.6745s+0.0029s)

To be completed...