On Erd\H{o}s constant

TL;DR

This paper investigates the Erdős constant by analyzing the growth of highly composite numbers, suggesting that the constant c is likely less than 1 based on numerical evidence.

Contribution

It provides numerical evidence indicating that the Erdős constant c is probably less than 1, refining previous theoretical bounds.

Findings

Numerical results suggest c<1 for the Erdős constant

Most large HCNs follow the growth pattern with c<1

Refines understanding of the distribution of highly composite numbers

Abstract

In 1944, P. Erd\H{o}s \cite{1} proved that if $n$ is a large highly composite number (HCN) and $n_1$ is the next HCN, then $$n<n_1<n+n(\log n)^{-c},$$ where $c>0$ is a constant. In this paper, using numerical results by D. A. Corneth, we show that most likely $c<1.$