Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves

TL;DR

This paper investigates the gaps between special integers called Hall numbers related to Marshall Hall's conjecture, establishing lower bounds on their differences and implications for points on Mordell elliptic curves.

Contribution

It proves new lower bounds on the gaps between Hall numbers, advancing understanding of their distribution and implications for elliptic curve points.

Findings

Proves that consecutive Hall numbers differ by more than (1/5) times their size to the 1/6 power.

Stronger gap bounds are established when Hall numbers are near perfect squares.

Results imply minimum distances between points on Mordell elliptic curves with large x-coordinates.

Abstract

For a non-square positive integer x, let k_x denote the distance between x^3 and the perfect square closest to x^3. A conjecture of Marshall Hall states that the ratios r_x = (x^(1/2))/k_x, are bounded above. (Elkies has shown that any such bound must exceed 46.6.) Let {x(n)} be the sequence of "Hall numbers": positive non-square integers for which r_x(n) exceeds 1. Extensive computer searches have identified approximately 50 Hall numbers. (It can be proved that infinitely many exist.) In this paper we study the minimum gap between consecutive Hall numbers. We prove that for all n, x(n + 1) - x(n) > (1/5)x(n)^(1/6), with stronger gaps applying when x(n) is close to perfect even or odd squares (approximately x(n)^(1/3) or x(n)^(1/4), respectively). This result has obvious implications for the minimum "horizontal gap" (and hence straight line and arc distance) between integer points (whose x-coordinates exceed k^2) on the Mordell elliptic curves x^3 - y^2 = k, a question that does not appear to have been addressed.