Found with PFGW Version 3.7.5

4433^8838-8838^4433 is 2-PRP! (18.1524s+0.0015s)
4433^8838-8838^4433 is 3-PRP! (18.0816s+0.0014s)
4433^8838-8838^4433 is 5-PRP! (18.0774s+0.0014s)
4433^8838-8838^4433 is 7-PRP! (18.0183s+0.0015s)
4433^8838-8838^4433 is 11-PRP! (18.0057s+0.0013s)

Primality testing 4433^8838-8838^4433 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 10+sqrt(13)
Calling N-1 BLS with factored part 0.04% and helper 0.00% (0.13% proof)
4433^8838-8838^4433 is Fermat and Lucas PRP! (111.5197s+0.0023s)

To be completed...