Right triangles with algebraic sides and elliptic curves over number fields-Reference-Cited by-同舟云学术

Right triangles with algebraic sides and elliptic curves over number fields

Published:2009-05-17 Issue:3 Volume:59 Page:299-306
ISSN:1337-2211
Container-title:Mathematica Slovaca
language:en
Short-container-title:

Author:

Girondo E.,González-Diez G.¹,González-Jiménez E.,Steuding R.²,Steuding J.²

Affiliation:

1. Departamento de Matemáticas, Universidad Autónoma de Madrid, C. Universitaria de Cantoblanco, ES-28 049, Madrid, Spain

2. Institut für Mathematik, Universität Würzburg, Am Hubland, DE-97 218, Würzburg, Germany

Abstract

Abstract Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

http://link.springer.com/content/pdf/10.2478/s12175-009-0126-3.pdf

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