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

TL;DR

This paper confirms a conjecture by Erdős and Pomerance by proving the existence of intervals of specific length that contain no distinct multiples of the first n positive integers, advancing understanding of number distribution.

Contribution

It proves the existence of intervals of length proportional to n log n / log log n that exclude all multiples of the first n integers, confirming a longstanding conjecture.

Findings

Intervals of length c n log n / log log n can avoid containing multiples of 1 to n

The result confirms a conjecture by Erdős and Pomerance

Advances understanding of the distribution of multiples in 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 $1, 2, \ldots, n$.