Constructing congruent number elliptic curves using 2-descent

2020/06/15 by Raiza Corpuz, Corpuz, Raiza
Mathematics · #11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.08113

openalex publication_date 2020/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that if n is a congruent number, then the equation nm² = uv(u² - v²) has a solution in ℤ. Using the relation between congruent numbers and elliptic curves E_n: y² = x³ - n² x which was established in the 1900s, we will prove that the converse of this two century-old result holds as well. In addition to this, we present another proof of the converse using the method of 2-descent. Towards the end of this paper, we demonstrate how one can use our proof to construct subfamilies of E_n with rank at least 2 and 3.

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