Three consecutive almost squares

Abstract

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