A short note on reduced residues

TL;DR

This paper investigates the behavior of the longest arithmetic progression within the least reduced residue system modulo n as n grows large, providing insights into its lower bound characteristics.

Contribution

It addresses Recamán's problem by establishing the lower bound behavior of maximum arithmetic progression lengths in reduced residue systems as n approaches infinity.

Findings

Established lower bound behavior for maximum arithmetic progression length

Extended understanding of residue system structures for large n

Provided theoretical insights into residue system progressions

Abstract

We solve a problem due to Recam\'an about the lower bound behavior of the maximum possible length among all arithmetic progressions in the least reduced residue system modulo $n$, as $n \to \infty$.