Numbers with three close factorizations and central lattice points on hyperbolas

TL;DR

This paper investigates the relationship between integers with three close factorizations and lattice points near the hyperbola's center, providing bounds and systematic examples using polynomials and Pell equations.

Contribution

It corrects a previous mistake and establishes optimal bounds for lattice points close to the hyperbola center, linking factorizations to geometric properties.

Findings

Established optimal lower bounds for L^1-distance to hyperbola center

Corrected a previous result regarding close factorizations

Provided systematic examples using polynomials and Pell equations

Abstract

In this paper, we continue the study of three close factorizations of an integer and correct a mistake of a previous result. This turns out to be related to lattice points close to the center point $(\sqrt{N}, \sqrt{N})$ of the hyperbola $x y = N$. We establish optimal lower bounds for $L^1$-distance between these lattice points and the center. We also give some good examples based on polynomials and Pell equations more systematically.