The diophantine equation $a{x}^{2}+bxy+c{y}^{2}=N$, $D={b}^{2}-4ac>0$
Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 257-270.

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$\left(a,b,c\right)=\text{pgcd}\left(a,N\right)=1\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}D={b}^{2}-4ac>0$ n’est pas un carré.

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$\left(a,b,c\right)=\text{gcd}\left(a,N\right)=1\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}D={b}^{2}-4ac>0$ 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 $N=±\mu$, where $\mu =mi{n}_{\left(x,y\right)\ne \left(0,0\right)}\left|a{x}^{2}+bxy+c{y}^{2}\right|$. We only need the special case $\mu =1$ of his result and give a self-contained proof, using our unimodular matrix approach.

@article{JTNB_2002__14_1_257_0,
author = {Matthews, Keith},
title = {The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {257--270},
publisher = {Universit\'e Bordeaux I},
volume = {14},
number = {1},
year = {2002},
doi = {10.5802/jtnb.358},
zbl = {1018.11013},
mrnumber = {1926002},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.358/}
}
Keith Matthews. The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$. Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 257-270. doi : 10.5802/jtnb.358. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.358/

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