Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the solution of $x^2-dy^2=\pm m$

Julius M. Basilla and Hideo Wada

Full-text: Open access


An improvement of the Gauss' algorithm for solving the diophantine equation $x^2-dy^2=\pm m$ is presented. As an application, multiple continued fraction method is proposed.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 8 (2005), 137-140.

First available in Project Euclid: 1 November 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations 11Y05: Factorization 11Y16: Algorithms; complexity [See also 68Q25]

Quadratic form diophantine equation continued fraction method prime decomposition


Basilla, Julius M.; Wada, Hideo. On the solution of $x^2-dy^2=\pm m$. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 8, 137--140. doi:10.3792/pjaa.81.137.

Export citation


  • J.M. Basilla, On the solution of $x^{2}+dy^{2}=m$, Proc. Japan Acad., 80A (2004), no. 5, 40–41.
  • H. Cohen, A course in computational algebraic number theory, Springer, Berlin, 1993.
  • C.F. Gauss, Disquisiones Arithmeticae, Fleischer, Leipzig, 1801.
  • L.K. Hua, Introduction to number theory, Translated from the Chinese by Peter Shiu, Springer, Berlin, 1982.
  • M.A. Morrison and J. Brillhart, A method of factoring and the factorization of $F_{7}$, Math. Comp. 29 (1975), 183–205.
  • O. Perron, Die Lehre von dem Kettenbrüchen I, Teubner, Stuttgart, 1954.
  • T. Takagi, Lectures on the elementary theory of numbers, 2nd ed., Kyoritsu-publication, Tokyo, 1971. (In Japanese).
  • H. Wada, A note on the Pell equation, Tokyo J. Math. 2 (1979), no. 1, 133–136.