Three consecutive almost squares

Abstract

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

arXiv commentMost difficult case now done using the Heegner point method