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Resolution of an Erdős' problem on least common multiples

Stijn Cambie

Abstract

Erdős posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question affirmatively by proving that the ratio of the lcm's can be made arbitrarily large.

Resolution of an Erdős' problem on least common multiples

Abstract

Erdős posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question affirmatively by proving that the ratio of the lcm's can be made arbitrarily large.

Paper Structure

This paper contains 3 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thr:main']}
  3. \ref{['ques:sieve-styleQ']} implies \ref{['conj:main2']}

Key Result

Theorem 1

Let $C \ge 1$ be a constant. For every $k$ sufficiently large, there exist integers $0<x<y$ with $y>x+k$ such that $\mathop{\mathrm{lcm}}\nolimits\{x,x+1,\ldots,x+k-1\} > C \cdot\mathop{\mathrm{lcm}}\nolimits\{y,y+1,\ldots,y+k\}.$

Theorems & Definitions (7)

  • Theorem 1
  • Conjecture 2
  • Claim 4
  • proof : Proof
  • Claim 5
  • proof : Proof
  • proof : Proof of \ref{['ques:sieve-styleQ']} $\Rightarrow$ \ref{['conj:main2']}