Coprime matchings

TL;DR

This paper proves the existence of coprime matchings between two intervals of integers under certain size conditions, improving previous results and contributing to the lonely runner conjecture.

Contribution

It establishes a new bound for coprime matchings between intervals, advancing understanding in combinatorial number theory.

Findings

Matchings exist for intervals longer than c(log n)^2

Improves previous bounds by Bohman and Peng

Application to the lonely runner conjecture

Abstract

We prove that there is a matching between 2 intervals of positive integers of the same even length, with corresponding pairs coprime, provided the intervals are in $[n]$ and their lengths are $>c(\log n)^2$, for a positive constant $c$. This improves on a recent result of Bohman and Peng. As in their paper, the result has an application to the lonely runner conjecture.