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

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.

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.

@article{JTNB_2002__14_1_257_0,
author = {Keith Matthews},
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},
zbl = {1018.11013},
mrnumber = {1926002},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_257_0/}
}
TY  - JOUR
AU  - Keith Matthews
TI  - The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
DA  - 2002///
SP  - 257
EP  - 270
VL  - 14
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_257_0/
UR  - https://zbmath.org/?q=an%3A1018.11013
UR  - https://www.ams.org/mathscinet-getitem?mr=1926002
LA  - en
ID  - JTNB_2002__14_1_257_0
ER  - 
%0 Journal Article
%A Keith Matthews
%T The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 257-270
%V 14
%N 1
%I Université Bordeaux I
%G en
%F JTNB_2002__14_1_257_0
Keith Matthews. The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 257-270. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_257_0/

[1] G. Cornacchia, Su di un metodo per la risoluxione in numeri interi dell' equazione ${\sum }_{h=0}^{n}{C}_{h}{x}^{n-h}=P$. Giornale di Matematiche di Battaglini 46 (1908), 33-90. | JFM

[2] A. Faisant, L 'equation diophantienne du second degré. Hermann, Paris, 1991. | MR | Zbl

[3] C.F. Gauss, Disquisitiones Arithmeticae. Yale University Press, New Haven, 1966. | MR | Zbl

[4] G.H. Hardy, E.M. Wright, An Introduction to Theory of Numbers, Oxford University Press, 1962. | MR

[5] L.K. Hua, Introduction to Number Theory. Springer, Berlin, 1982. | MR | Zbl

[6] G.B. Mathews, Theory of numbers, 2nd ed. Chelsea Publishing Co., New York, 1961. | JFM | MR

[7] K.R. Matthews, The Diophantine equation x2 - Dy2 = N, D > 0. Exposition. Math. 18 (2000), 323-331. | MR | Zbl

[8] R.A. Mollin, Fundamental Number Theory with Applications. CRC Press, New York, 1998. | Zbl

[9] A. Nitaj, Conséquences et aspects expérimentaux des conjectures abc et de Szpiro. Thèse, Caen, 1994.

[10] M. Pavone, A Remark on a Theorem of Serret. J. Number Theory 23 (1986), 268-278. | MR | Zbl

[11] J. A. SERRET (Ed.), Oeuvres de Lagrange, I-XIV, Gauthiers-Villars, Paris, 1877.

[12] J.A. Serret, Cours d'algèbre supérieure, Vol. I, 4th ed. Gauthiers-Villars, Paris, 1877. | JFM

[13] T. Skolem, Diophantische Gleichungen, Chelsea Publishing Co., New York, 1950.