Right triangles with algebraic sides and elliptic curves over number fields
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 positiv...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2009 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/711109 |
| Acceso en línea: | http://hdl.handle.net/10486/711109 https://dx.doi.org/10.2478/s12175-009-0126-3 |
| Access Level: | acceso abierto |
| Palabra clave: | Congruent Number Problem Elliptic Curves Number Fields Matemáticas |
| Sumario: | 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). © 2009 © Versita Warsaw and Springer-Verlag Berlin Heidelberg |
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