Distance between arithmetic progressions and perfect squares

TL;DR

This paper investigates the proximity of finite arithmetic progression terms to perfect squares, analyzing how initial term, common difference, and number of terms influence this closeness.

Contribution

It provides new bounds and insights into the minimal distance between arithmetic progression terms and perfect squares based on progression parameters.

Findings

Derived bounds for the minimal distance to perfect squares

Identified conditions under which progression terms are close to squares

Extended previous results on the distribution of squares in progressions

Abstract

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.