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Even after a lot of search on internet, I didn't come up with a solution. I need an algorithm to efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$.

Lattice point means integer coordinates and equation with integer means diophantine equations. The most closed resources I've found are:

This Link and this other post.

But the former look for solutions of equations and the latter is asking something linked but different.

So, there is an algorithm (better if efficient of course) or this is an unsolved/open problem?

Servaes
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user2548436
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1 Answers1

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Given integers $a$, $b$, $c$ and $d$ we want to find a pairs of integer $(x,y)$ such that $$axy+bx+cy+d=0.$$ Note that if $(x,y)$ is such a pair, then \begin{eqnarray*} (ax+c)(ay+b)&=& a^2xy+abx+acy+bc\\ &=&a(axy+bx+cy+d)-ad+bc\\ &=&bc-ad. \end{eqnarray*} Let $N:=bc-ad$. This shows that lattice points on the hyperbola correspond bijectively to factorizations $N=uv$ with $u\equiv b\pmod{a}$ and $v\equiv c\pmod{a}$. Of course factorization methods have been studied plenty, and this particular factorization problem may allow for some slightly more effective methods than for general factorization of integers.

Note that such factorizations need not exist; for example $$5xy+3x+2y+1=0,$$ has no integral solutions, because $N=bc-ad=1$ has no factors congruent to $2$ or $3$ mod $5$.

Servaes
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