Rational points on elliptic curves and representations of rational numbers as the product of two rational factors

TL;DR

This paper explores the limited ways rational numbers can be factored to share sum properties and examines elliptic curves with rational roots that exhibit these factorization characteristics.

Contribution

It introduces a novel bound on the number of rational factorizations with shared sum properties and analyzes elliptic curves with specific rational root conditions.

Findings

Maximum of four factorizations with shared sum properties

Characterization of elliptic curves with rational roots satisfying the property

Insights into the structure of rational points on elliptic curves

Abstract

In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of n. In the second part of the paper I derive the properties of elliptic curves such that the cubic has rational roots that satisfy this property