# On the distribution of consecutive square-free numbers of the form $\mathbf{[\alpha n], [\alpha n]+1}$

## TL;DR

This paper proves the existence of infinitely many consecutive square-free numbers of specific forms involving irrational numbers with certain properties, expanding understanding of their distribution.

## Contribution

It establishes the infinite occurrence of consecutive square-free numbers of the form [αn], [αn]+1 for irrational α with bounded partial quotients or algebraic irrationals.

## Key findings

- Infinitely many such pairs exist for specified irrational α.
- The result applies to α with bounded partial quotients or algebraic irrationals.
- The proof involves advanced number theory techniques.

## Abstract

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[\alpha n]$, $[\alpha n]+1$, where $\alpha>1$ is irrational number with bounded partial quotient or irrational algebraic number.