A Generalization of Siegel's Theorem and Hall's Conjecture

TL;DR

This paper explores a theoretical and computational extension of Siegel's Theorem and Hall's conjecture, focusing on rational points on elliptic curves with bounded prime factors in their coordinates.

Contribution

It introduces generalized conjectures extending classical results, providing new insights into the distribution of rational points with restricted prime factors.

Findings

Finite points for zero prime factors bound confirmed by Siegel's Theorem

Proposes conjectures for bounded prime factors in rational points

Computational evidence supporting the generalized conjectures

Abstract

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a fixed coordinate? If the bound is zero, then Siegel's Theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel's Theorem and the other is a refinement which resonates with Hall's conjecture.