Found with PFGW Version 3.7.5

1223^8844-8844^1223 is 2-PRP! (14.0806s+0.0013s)
1223^8844-8844^1223 is 3-PRP! (14.0265s+0.0013s)
1223^8844-8844^1223 is 5-PRP! (14.3037s+0.0011s)
1223^8844-8844^1223 is 7-PRP! (14.0608s+0.0012s)
1223^8844-8844^1223 is 11-PRP! (14.6190s+0.0010s)

Primality testing 1223^8844-8844^1223 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 7
Running N-1 test using base 17
Running N-1 test using base 19
Running N-1 test using base 29
Running N+1 test using discriminant 41, base 1+sqrt(41)
Calling N-1 BLS with factored part 0.03% and helper 0.00% (0.09% proof)
1223^8844-8844^1223 is Fermat and Lucas PRP! (126.2100s+0.0019s)

To be completed...