On coprimality of consecutive elements in certain sequences
Jean-Marc Deshouillers, Sunil Naik

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 is a twice continuously differentiable real-valued function on such that as and , then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence . 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 having upper Banach density one such that for any two distinct integers , the integers and $\lfloor f(n)…
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Taxonomy
TopicsAnalytic Number Theory Research · Limits and Structures in Graph Theory · Advanced Banach Space Theory
