Integral solutions of quadratic equation $5 X² − 14 X⁢Y + 5 ⁢Y² = n$

Question

Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 X⁢Y + 5 ⁢Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these sorts of equations in general. Can someone lend help or a good in depth resource for these sorts of problems?

Answer 1

Conway's construction of the topograph of a quadratic form $ax^2 + bxy +cy^2$ gives a way to find all integer solutions of an equation $ax^2 + bxy +cy^2=n$ since the topograph displays all the solutions with $x$ and $y$ coprime, for all $n$. The form $5x^2-14xy+5y^2$ has positive nonsquare discriminant so its topograph is periodic under a matrix in $SL(2,{\mathbb Z})$, hence it suffices to find all of the finitely many solutions of $5x^2-14xy+5y^2=n$ in one period and then apply all positive and negative powers of the periodicity matrix to these solutions. From the topograph one can read off the periodicity matrix, which for the form $5x^2-14xy+5y^2$ is the matrix with rows $(2,-5)$ and $(5,-12)$. Thus each solution $(x,y)$ gives another solution $(2x-5y,5x-12y)$. Similarly, using the inverse of the periodicity matrix one gets $(-12x+5y,-5x+2y)$. (This particular form has symmetry with respect to interchanging $x$ and $y$ so there is no need to consider the inverse of the periodicity matrix and one can just switch $x$ and $y$ instead.)

For example for $n=5$ one sees from the topograph that the only solutions in one period are $(x,y)=\pm(1,0)$ and $\pm(0,1)$. These yield two infinite sequences of solutions $\pm(x,y)$ by applying the recurrence $(x,y)\mapsto(2x-5y,5x-12y)$ repeatedly, and all the remaining solutions come from interchanging $x$ and $y$.

One can find an introduction to topographs in Conway's book "The Sensual Quadratic Form". For a more detailed exposition see my book "Topology of Numbers" which is due to be published in November 2022 by the AMS, with a free electronic version available permanently on my webpage.

Answer 2

One approach is to multiply through by $5$ and complete the square to rewrite the equation as $U^2-6V^2=5n$ where $U = 5X-7Y$ and $V=2Y$.

An example of finding all integral solutions to $x^2-6y^2=m$ for some $m$ is Example 4.1 here.