Right triangles with algebraic sides and elliptic curves over number fields

TL;DR

This paper proves the existence of infinitely many right triangles with a given area and sides in specific number fields, generalizing the congruent number problem and providing explicit constructions via elliptic curves.

Contribution

It introduces a method to construct such triangles explicitly using elliptic curves over specially chosen cubic number fields, extending classical results.

Findings

Existence of infinitely many right triangles with given area in certain number fields.

Explicit construction of these triangles using elliptic curves.

Identification of specific cubic number fields for each positive integer n.

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).