The number of solutions to Mordell's equations in constrained ranges

TL;DR

This paper estimates the quantity of solutions to Mordell's equations within specific bounds, relating the number of solutions to parameters N and X, advancing understanding of their distribution.

Contribution

It provides new bounds and estimates for the number of solutions to Mordell's equations in constrained ranges, connecting solution counts to parameters N and X.

Findings

Derived bounds for solutions in specified ranges

Established relationships between N, X, and solution counts

Enhanced understanding of Mordell's equations solutions distribution

Abstract

We estimate the number of solutions to $|y^2 - x^3| \leq X$ with $N \leq y \leq 2N$, in terms of both $N$ and $X$.