Found with PFGW Version 3.6.0

844^6237-6237^844 is 2-PRP! (14.0273s+0.0020s)
844^6237-6237^844 is 3-PRP! (13.9223s+0.0020s)
844^6237-6237^844 is 5-PRP! (13.9787s+0.0021s)
844^6237-6237^844 is 7-PRP! (14.0089s+0.0023s)
844^6237-6237^844 is 11-PRP! (13.8714s+0.0021s)

Primality testing 844^6237-6237^844 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 37, base 1+sqrt(37)
Calling N-1 BLS with factored part 0.07% and helper 0.01% (0.21% proof)
844^6237-6237^844 is Fermat and Lucas PRP! (81.1679s+0.0020s)

To be completed...