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

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

Julius Magalona Basilla

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

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.

Primary Subjects: 11Y16

Secondary Subjects: 11D09

Keywords: Cornacchia's algorithm; quadratic forms; diophantine equations

Full-text: Open access

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Links and Identifiers

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

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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).

Mathematical Reviews (MathSciNet): MR1228206

Zentralblatt MATH: 0786.11071

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).

Mathematical Reviews (MathSciNet): MR541903

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