Is there a general solution to solve a Diophantine equation of the form $Axy + Bx + Cy + D = N$?
With $A,B,C,D,N,x,y$ positive integers.
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Do you allow negative integers? – Somnium Apr 30 '17 at 09:35
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No, all A,B,C,D,N,x,y should positive integers. I'm updating the question. – Augusto Altman Quaranta Apr 30 '17 at 09:37
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Please refrain from duplicating @Lord Shark' s answer. I found two copies after I posted my own answer, my own being one of them, and thus I deleted my version and gave that answerer +1. – Oscar Lanzi Apr 30 '17 at 09:58
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We have $$AN=A^2xy+ABx+ACy+AD=(Ax+C)(Ay+B)+AD-BC.$$ This reduces to $$uv=K$$ where $K=AD-BC-AN$ is known and $u=Ax+C$ and $v=Ay+B$. So we are looking for factorisations of $K$ with extra conditions $u\equiv C\pmod A$ and $v\equiv B\pmod A$.
Angina Seng
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It's factorization (as done by LStU) is known as :
Simon's Favorite Factorization Trick
The general statement of SFFT is $$xy+yj+ xk+jk =(x+j)(y+k)$$ The act of adding $jk$ to $xy +xk+yj$ in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."
Now you can prime factorize the constant term in order to get solutions of $x$ and $y$.
Jaideep Khare
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