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

• Volume: 14, Issue: 1, page 257-270
• ISSN: 1246-7405

top

## Abstract

top
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$\left(a,b,c\right)=\text{gcd}\left(a,N\right)=1\phantom{\rule{4.0pt}{0ex}}\text{et}\phantom{\rule{4.0pt}{0ex}}D={b}^{2}-4ac>0$ 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.

## How to cite

top

Matthews, Keith. "The diophantine equation $ax^2 + bxy + cy^2 = N, D = b^2 - 4ac &gt; 0$." Journal de théorie des nombres de Bordeaux 14.1 (2002): 257-270. <http://eudml.org/doc/248916>.

@article{Matthews2002,
abstract = {We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $ax^2 + bxy + cy^2 = N$ in relatively prime integers $x, y$, where $N \ne 0$, gcd$(a, b, c) = \text\{gcd\}(a, N) = 1 \text\{ et \} D = b^2 - 4ac &gt; 0$ 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 $x^2 - Dy^2 = N$. 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 $\mu = min_\{(x,y) \ne (0,0)\} \left|ax^2+bxy+cy^2\right|$. We only need the special case $\mu =1$ of his result and give a self-contained proof, using our unimodular matrix approach.},
author = {Matthews, Keith},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {quadratic Diophantine equation; continued fraction; unimodular matrix},
language = {eng},
number = {1},
pages = {257-270},
publisher = {Université Bordeaux I},
title = {The diophantine equation $ax^2 + bxy + cy^2 = N, D = b^2 - 4ac &gt; 0$},
url = {http://eudml.org/doc/248916},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Matthews, Keith
TI - The diophantine equation $ax^2 + bxy + cy^2 = N, D = b^2 - 4ac &gt; 0$
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 257
EP - 270
AB - We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $ax^2 + bxy + cy^2 = N$ in relatively prime integers $x, y$, where $N \ne 0$, gcd$(a, b, c) = \text{gcd}(a, N) = 1 \text{ et } D = b^2 - 4ac &gt; 0$ 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 $x^2 - Dy^2 = N$. 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 $\mu = min_{(x,y) \ne (0,0)} \left|ax^2+bxy+cy^2\right|$. We only need the special case $\mu =1$ of his result and give a self-contained proof, using our unimodular matrix approach.
LA - eng
KW - quadratic Diophantine equation; continued fraction; unimodular matrix
UR - http://eudml.org/doc/248916
ER -

## References

top
1. [1] G. Cornacchia, . Giornale di Matematiche di Battaglini46 (1908), 33-90. JFM39.0258.02
2. [2] A. Faisant, L 'equation diophantienne du second degré. Hermann, Paris, 1991. Zbl0757.11012MR1169678
3. [3] C.F. Gauss, Disquisitiones Arithmeticae. Yale University Press, New Haven, 1966. Zbl0136.32301MR197380
4. [4] G.H. Hardy, E.M. Wright, An Introduction to Theory of Numbers, Oxford University Press, 1962. Zbl0020.29201MR568909
5. [5] L.K. Hua, Introduction to Number Theory. Springer, Berlin, 1982. Zbl0483.10001MR665428
6. [6] G.B. Mathews, Theory of numbers, 2nd ed. Chelsea Publishing Co., New York, 1961. MR126402JFM24.0162.01
7. [7] K.R. Matthews, The Diophantine equation x2 - Dy2 = N, D &gt; 0. Exposition. Math.18 (2000), 323-331. Zbl0976.11016MR1788328
8. [8] R.A. Mollin, Fundamental Number Theory with Applications. CRC Press, New York, 1998. Zbl0943.11001
9. [9] A. Nitaj, Conséquences et aspects expérimentaux des conjectures abc et de Szpiro. Thèse, Caen, 1994.
10. [10] M. Pavone, A Remark on a Theorem of Serret. J. Number Theory23 (1986), 268-278. Zbl0588.10020MR845908
11. [11] J. A. SERRET (Ed.), Oeuvres de Lagrange, I-XIV, Gauthiers-Villars, Paris, 1877.
12. [12] J.A. Serret, Cours d'algèbre supérieure, Vol. I, 4th ed. Gauthiers-Villars, Paris, 1877. Zbl54.0117.01JFM17.0053.01
13. [13] T. Skolem, Diophantische Gleichungen, Chelsea Publishing Co., New York, 1950.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.