Right triangles with algebraic sides and elliptic curves over number fields
Number Theory
2014-11-14 v1
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(\lambda) (depending on n) and an explicit point P_\lambda of infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(\lambda).
Cite
@article{arxiv.0903.4611,
title = {Right triangles with algebraic sides and elliptic curves over number fields},
author = {Ernesto Girondo and Gabino Gonzalez-Diez and Enrique Gonzalez-Jimenez and Rasa Steuding and Jorn Steuding},
journal= {arXiv preprint arXiv:0903.4611},
year = {2014}
}
Comments
To appear in Math. Slovaca