Consecutive square-free values of the form $\mathbf{[\alpha p], [\alpha p]+1}$

Abstract

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[\alpha p]$, $[\alpha p]+1$, where $p$ is prime and $\alpha>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive integer.