Resolution of an Erd\H{o}s' problem on least common multiples

TL;DR

This paper proves that there are infinitely many sets of consecutive numbers whose least common multiples can be made arbitrarily large compared to larger sets, resolving a longstanding question posed by Erdős.

Contribution

The paper provides a proof that the ratio of the least common multiples of certain consecutive sets can be arbitrarily large, answering Erdős's problem.

Findings

Existence of infinitely many sets with arbitrarily large lcm ratios

Construction method for such sets

Resolution of Erdős's open problem

Abstract

Erd\H{o}s 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.