Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the solution of $x^2 + dy^2 = m$

Julius Magalona Basilla

Full-text: Open access

Abstract

A simple proof of the validity of Cornacchia's algorithm for solving the diophantine equation $x^2 + dy^2 = m$ is presented. Furthermore, the special case $d=1$ is solved completely.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 80, Number 5 (2004), 40-41.

Dates
First available: 18 May 2005

Permanent link to this document
http://projecteuclid.org/euclid.pja/1116442240

Mathematical Reviews number (MathSciNet)
MR2062797

Zentralblatt MATH identifier
02138333

Digital Object Identifier
doi:10.3792/pjaa.80.40

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

Keywords
Cornacchia's algorithm quadratic forms diophantine equations

Citation

Basilla, Julius Magalona. On the solution of $x^2 + dy^2 = m$. Proceedings of the Japan Academy, Series A, Mathematical Sciences 80 (2004), no. 5, 40--41. doi:10.3792/pjaa.80.40. http://projecteuclid.org/euclid.pja/1116442240.


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References

  • Cohen, H.: A Course in Computational Number Theory. Grad. Texts in Math., 138. Springer-Verlag, New York, pp. 34–36 (1993).
  • Morain, F., and Nicolas, J.-L.: On Cornacchia's Algorithm for Solving the Diophantine Equation $u^2+dv^2=m$. Courbes elliptiques et tests de primalite These, Universite de Lyon I, 20 September (1990).
  • Wada, H.: A note on the Pell equation. Tokyo J. Math., 2, 133–136 (1979).