The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$

Matthews, Keith

The diophantine equation

a x^{2} + b x y + c y^{2} = N

D = b^{2} - 4 a c > 0

Matthews, Keith

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

Résumé
Abstract

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

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a x^{2} + b x y + c y^{2} = N$ in relatively prime integers $x, y$ , where $N \neq 0$ , gcd $(a, b, c) = gcd (a, N) = 1 et D = b^{2} - 4 a c > 0$ 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 $N = \pm μ$ , where $μ = m i n_{(x, y) \neq (0, 0)} |a x^{2} + b x y + c y^{2}|$ . We only need the special case $μ = 1$ of his result and give a self-contained proof, using our unimodular matrix approach.

MR Zbl

@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},
     mrnumber = {1926002},
     zbl = {1018.11013},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_1_257_0/}
}

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Matthews, Keith. 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. http://www.numdam.org/item/JTNB_2002__14_1_257_0/

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