The diophantine equation a x 2 + b x y + c y 2 = N , D = b 2 - 4 a c > 0

Keith Matthews

Journal de théorie des nombres de Bordeaux (2002)

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

Abstract

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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 ≠ 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 = ± μ , where μ = m i n ( x , y ) ≠ ( 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.

How to cite

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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

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  2. [2] A. Faisant, L 'equation diophantienne du second degré. Hermann, Paris, 1991. Zbl0757.11012MR1169678
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  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. 

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