An identity involving the least common multiple of binomial coefficients and its application

TL;DR

This paper proves a new identity relating the least common multiple of binomial coefficients to the LCM of integers and uses it to provide a simple proof of a known lower bound for the LCM of the first k natural numbers.

Contribution

It introduces a novel identity connecting binomial coefficient LCMs with integer LCMs and applies it to simplify the proof of a classical lower bound.

Findings

Established the identity: lcm of binomial coefficients equals lcm of integers divided by (k+1).

Provided a straightforward proof of the lower bound lcm(1,2,...,k) ≥ 2^{k-1}.

Enhanced understanding of the relationship between binomial coefficients and number theory.

Abstract

In this paper, we prove the identity $$\lcm\{\binom{k}{0}, \binom{k}{1}, >..., \binom{k}{k}\} = \frac{\lcm(1, 2, ..., k, k + 1)}{k + 1} (\forall k \in \mathbb{N}) .$$ As an application, we give an easily proof of the well-known nontrivial lower bound $\lcm(1, 2, ..., k) \geq 2^{k - 1}$ $(\forall k \geq 1)$.