Consecutive square-free values of the form [αp], [αp]+1

2019/07/03 by S. I. Dimitrov, Dimitrov, S. I.
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1907.03721

openalex publication_date 2019/07/03 · openalex created_date 2019/07/12 · openalex updated_date 2026/07/28

Abstract

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form [αp], [αp]+1, where p is prime and α>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.

Related

• On the distribution of consecutive square-free numbers of the form [αn], [αn]+1
• Pairs of square-free values of the type n²+1, n²+2
• Consecutive square-free numbers of a special form
• On the number of pairs of positive integers x, y ≤ H such that x²+y²+1, x²+y²+2 are square-free
• Consecutive square-free values for some polynomials