Three consecutive near-square squarefree numbers

TL;DR

This paper proves there are infinitely many integers n such that n^2+1, n^2+2, and n^2+3 are all squarefree, and provides an asymptotic formula for their distribution.

Contribution

It establishes the infinitude of such triplets and refines the error term to derive an asymptotic formula for their count.

Findings

Infinitely many n with all three numbers squarefree

An asymptotic formula for the count of such n

Improved error term in related cases

Abstract

In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error term slightly in the case of two consecutive numbers of the same form, so that we are able to prove the following asymptotic formula. \begin{align*} \sum_{n\le X}\mu^2(n^2+1)\mu^2(n^2+2)\mu^2(n^2+3)\sim\dfrac{7}{18}\prod_{p>3}\left(1-\dfrac{3+\left(\frac{-1}{p}\right)+\left(\frac{-2}{p}\right)+\left(\frac{-3}{p}\right)}{p^2}\right)X. \end{align*}