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On the length of an interval that contains distinct multiples of the first $n$ positive integers

Wouter van Doorn

TL;DR

The paper settles the Erdős-Pomerance conjecture by showing there exist intervals of length $\frac{0.36\,n\log n}{\log\log n}$ that do not contain distinct multiples of $1,\dots,n$, thereby establishing a nontrivial growth rate for gaps between such multiples. The proof hinges on a fundamental inequality $kn+f(kn,kn)\le k^2n+f(n,k^2n)$, together with the best-known bounds for $f(n,n)$, and a carefully chosen parameter $k=\lceil 0.6\sqrt{\frac{\log n}{\log\log n}}\rceil$ to separate $f(n,n)$ from larger $f(n,m)$. This yields a positive lower bound on $\max_m f(n,m)-f(n,n)$, confirming the conjecture and advancing our understanding of divisibility-structure gaps in intervals. The result provides a concrete asymptotic rate for constructing long intervals void of distinct multiples of the first $n$ integers and contributes to extremal problems in additive number theory.

Abstract

Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length $\frac{cn\log n}{\log \log n}$ that do not contain distinct multiples of $1, 2, \ldots, n$.

On the length of an interval that contains distinct multiples of the first $n$ positive integers

TL;DR

The paper settles the Erdős-Pomerance conjecture by showing there exist intervals of length that do not contain distinct multiples of , thereby establishing a nontrivial growth rate for gaps between such multiples. The proof hinges on a fundamental inequality , together with the best-known bounds for , and a carefully chosen parameter to separate from larger . This yields a positive lower bound on , confirming the conjecture and advancing our understanding of divisibility-structure gaps in intervals. The result provides a concrete asymptotic rate for constructing long intervals void of distinct multiples of the first integers and contributes to extremal problems in additive number theory.

Abstract

Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length that do not contain distinct multiples of .
Paper Structure (2 sections, 3 theorems, 6 equations)

This paper contains 2 sections, 3 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1

We have the lower bound for all large enough $n \in \mathbb{N}$. In particular, if $n$ is sufficiently large, then an interval of length $\frac{0.36n \log n}{\log \log n}$ exists that does not contain distinct multiples of $1, 2, \ldots, n$.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof : Proof of Theorem \ref{['main']}