The question is : for integers $a,b,c,N$ with $N>0$, $a^2 + b^2 + c^2 - Nabc = 2022$. Find the $N$ where integer solutions of the equation exists.
I think we should make a secondary surplus using a modular. In addition, if one magnetic root exists, the solution will be infinite. Through the relationship between root and coefficient. I'd appreciate it if you answer me!
+ if N==1 or 2(mod 3), Because 2022==3(mod 9), solution doesn't exists.