Square-free values of $\mathbf{n^2+n+1}$

TL;DR

This paper proves there are infinitely many square-free numbers of the form n^2 + n + 1 by deriving an improved asymptotic formula with a better error term.

Contribution

It provides the first proof of infinitely many square-free values of n^2 + n + 1 with an improved asymptotic estimate.

Findings

Infinitely many square-free numbers of the form n^2 + n + 1 exist.

An improved asymptotic formula with a better error term was derived.

The result advances understanding of square-free values of quadratic polynomials.

Abstract

In this paper we show that there exist infinitely many square-free numbers of the form $n^2+n+1$. We achieve this by deriving an asymptotic formula by improving the reminder term from previous results.