Halfway to a solution of X 2 - D Y 2 = - 3

R. A. Mollin; A. J. Van der Poorten; H. C. Williams

Journal de théorie des nombres de Bordeaux (1994)

  • Volume: 6, Issue: 2, page 421-457
  • ISSN: 1246-7405

Abstract

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It is well known that the continued fraction expansion of D readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of x 2 - D y 2 = ± 1 . Here we notice that, analogously, the point halfway to a solution of x 2 - D y 2 = - 3 can be recognised. We explain what is going on.

How to cite

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Mollin, R. A., Van der Poorten, A. J., and Williams, H. C.. "Halfway to a solution of $X^2 - DY^2 = -3$." Journal de théorie des nombres de Bordeaux 6.2 (1994): 421-457. <http://eudml.org/doc/93612>.

@article{Mollin1994,
abstract = {It is well known that the continued fraction expansion of $\sqrt\{D\}$ readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of $x^2 - Dy^2 = \pm 1$. Here we notice that, analogously, the point halfway to a solution of $x^2 - Dy^2 = -3$ can be recognised. We explain what is going on.},
author = {Mollin, R. A., Van der Poorten, A. J., Williams, H. C.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {continued fraction; ideal; quadratic form; ambiguous cycle; quadratic diophantine equations; quadratic extensions; continued fraction expansion; principal cycle of ideals; exact calculations},
language = {eng},
number = {2},
pages = {421-457},
publisher = {Université Bordeaux I},
title = {Halfway to a solution of $X^2 - DY^2 = -3$},
url = {http://eudml.org/doc/93612},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Mollin, R. A.
AU - Van der Poorten, A. J.
AU - Williams, H. C.
TI - Halfway to a solution of $X^2 - DY^2 = -3$
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 2
SP - 421
EP - 457
AB - It is well known that the continued fraction expansion of $\sqrt{D}$ readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of $x^2 - Dy^2 = \pm 1$. Here we notice that, analogously, the point halfway to a solution of $x^2 - Dy^2 = -3$ can be recognised. We explain what is going on.
LA - eng
KW - continued fraction; ideal; quadratic form; ambiguous cycle; quadratic diophantine equations; quadratic extensions; continued fraction expansion; principal cycle of ideals; exact calculations
UR - http://eudml.org/doc/93612
ER -

References

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  1. [1] H.W. Lenstra Jr, On the calculation of regulators and class numbers of quadratic fields, J. V. ARMITAGE ed., Journées Arithmétiques 1980, LMS Lecture Notes56, Cambridge, 1982, pp. 123-151.. Zbl0487.12003MR697260
  2. [2] R.A. Mollin and A.J. Van Der Poorten, A note on symmetry and ambiguity, Bull. Austral. Math. Soc.51 (1995), 215-233. Zbl0824.11003MR1322789
  3. [3] Oskar Perron, Die Lehre von den Kettenbrüchen, (Chelsea reprint of 1929 edition). Zbl0056.05901
  4. [4] A.J. Van Der Poorten, An introduction to continued fractions, Diophantine Analysis, LMS Lecture Notes in Math.109, ed. J. H. LOXTON and A. J. VAN DER POORTEN, Cambridge University Press, 1986, pp. 99-138. Zbl0596.10008MR874123
  5. [5] A.J. Van Der Poorten, Fractions of the period of the continued fraction expansion of quadratic integers, Bull. Austral. Math. Soc44 (1991), 155-169. Zbl0733.11004MR1120403
  6. [6] D. Shanks, Class number, a theory of factorization, and genera, Proc. Symp. Pure Math., 20 (1969 Institute on Number Theory), Amer. Math. Soc., Providence1971, pp. 415-440, see also The infrastructure of a real quadratic field and its applications, Proc. Number Theory Conference, Boulder, 1972. Zbl0223.12006MR316385
  7. [7] D. Shanks, On Gauβ and composition, Number Theory and Applications, Richard A. MOLLIN ed. (NATO - Advanced Study Institute, Banff, 1988), Kluwer Academic Publishers, Dordrecht, 1989, pp. 163-204. 

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