The diophantine equation $a{x}^2+bxy+c{y}^2=N$, $D=b^2-4ac>0$

Matthews, Keith

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 257-270

Abstract (VO)

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a{x}^2+bxy+c{y}^2=N$ in relatively prime integers $x,y$, where $N\ne 0$, gcd$(a,b,c)=\text{gcd}(a,N)=1\text{et}D=b^2-4ac>0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation $x^2-D{y}^2=N$. As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for the necessity of the existence of a solution. Lagrange did not discuss an exceptional case which can arise when $D=5$. This was done by M. Pavone in 1986, when $N=\pm \mu$, where $\mu =mi{n}_{(x,y)\ne (0,0)}\left|a{x}^2+bxy+c{y}^2\right|$. We only need the special case $\mu =1$ of his result and give a self-contained proof, using our unimodular matrix approach.

Résumé

Nous revisitons un algorithme dû à Lagrange, basé sur le développement en fraction continue, pour résoudre l’équation $a{x}^2+bxy+c{y}^2=N$ en les entiers $x,y$ premiers entre eux, où $N\ne 0$, pgcd$(a,b,c)=\text{pgcd}(a,N)=1\text{et}D=b^2-4ac>0$ n’est pas un carré.