Etude du graphe divisoriel 4

TL;DR

This paper constructs a permutation of positive integers with controlled least common multiples between consecutive terms, improving bounds established in earlier research.

Contribution

It introduces a permutation of positive integers satisfying a tighter bound on the least common multiple of consecutive terms, refining previous results.

Findings

Existence of permutation with bounded l.c.m. for consecutive integers

Improved upper bound on l.c.m. compared to prior work

Advancement in understanding the structure of integer permutations

Abstract

We show that there is a permutation $f$ of the positive integers such that for $n \geq 2,$ l.c.m.$(f(n), f(n+1)) \leq cn(\log n)^2,$ where $c$ is a positive constant. It improves previous results of Erd\"os, Freud and Hegyvari (1983), and Chen and Ji (2011).