On coprimality of consecutive elements in certain sequences

TL;DR

This paper investigates the coprimality properties of consecutive elements in sequences derived from smooth functions, showing the existence of arbitrarily long coprime blocks and subsets with high density, refining previous results in number theory.

Contribution

It establishes new conditions under which sequences have long blocks of pairwise coprime elements and high-density subsets with this property, improving upon recent work.

Findings

Existence of arbitrarily long coprime blocks in certain sequences.

Construction of high-density subsets with pairwise coprimality.

Examples of blocks with no pairwise coprimality.

Abstract

The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$ and $\limsup_{x \to \infty} f'(x) = \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$. This result refines the qualitative part of a recent result by the first author, Drmota and M\"{u}llner. We also prove that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are pairwise coprime.