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

Journal de théorie des nombres de Bordeaux (2002)

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

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topMatthews, Keith. "The diophantine equation $ax^2 + bxy + cy^2 = N, D = b^2 - 4ac > 0$." 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$(a, b, c) = \text\{gcd\}(a, N) = 1 \text\{ et \} D = b^2 - 4ac > 0$ 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 $ax^2 + bxy + cy^2 = N, D = b^2 - 4ac > 0$},

url = {http://eudml.org/doc/248916},

volume = {14},

year = {2002},

}

TY - JOUR

AU - Matthews, Keith

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

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$(a, b, c) = \text{gcd}(a, N) = 1 \text{ et } D = b^2 - 4ac > 0$ 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

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- [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. Zbl54.0117.01JFM17.0053.01
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