Factors of almost squares and lattice points on circles

TL;DR

This paper investigates conjectures related to almost squares and lattice points on circles, using specific methods to analyze their properties and distributions near the x-axis for particular radii.

Contribution

It introduces new approaches to study conjectures of Erdos, Rosenfeld, and Ruzsa concerning almost squares and lattice points on circles.

Findings

Analysis of lattice points near the x-axis for special radii

Results supporting conjectures for almost squares

New methods for studying lattice point distributions

Abstract

In this paper, we consider a conjecture of Erdos and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.