Three consecutive almost squares

TL;DR

The paper characterizes positive integers with bounded squarefree parts in three consecutive numbers and proves infinitely many such n with a bound less than n^{1/3}, using elliptic curves.

Contribution

It provides a complete classification for bounded squarefree parts in three consecutive integers and establishes an infinite family with a specific growth bound.

Findings

All n with max squarefree part ≤ 150 are determined.

Infinitely many n have max squarefree part less than n^{1/3}.

The problem is linked to finding integral points on elliptic curves.

Abstract

Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many $n$ for which \[ \max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} < n^{1/3}. \]