TITLE: What is the time complexity for solving Diophantine equations of degree 2?
QUESTION [5 upvotes]: Manders and Adleman mention that the computational complexity for binary quadratic Diophantine equations is NP-complete. Has a more specific complexity been claimed for polynomials of the form $Axy + Bx + Cy = D$ where the coefficients are nonnegative integers? The only algorithm I have encountered so far is Alpern's method for solving $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, so I use that one and set $A=0, C=0$. I am looking for the most efficient algorithm (or the lowest complexity claim for such an algorithm) for solving this "simpler" form.

REPLY [2 votes]: $Axy+Bx+Cy=D$ is $(Ax+C)(Ay+B)=AD+BC$ so it essentially equivalent to factoring $AD+BC$.