Nair's and Farhi's identities involving the least common multiple of binomial coefficients are equivalent

TL;DR

This paper demonstrates that two recent identities involving the least common multiple of binomial coefficients, proved by Nair and Farhi, are mathematically equivalent, unifying their results.

Contribution

It establishes the equivalence between Nair's and Farhi's identities, providing a deeper understanding of the relationships among binomial coefficients and least common multiples.

Findings

Nair's and Farhi's identities are equivalent.

Unified understanding of binomial coefficient identities.

Clarifies the relationship between two recent mathematical results.

Abstract

In 1982, Nair proved the identity: ${\rm lcm}({k \choose 1}, 2{k \choose 2}, >..., k{k \choose k}) ={\rm lcm}(1, 2, ..., k), \forall k\in \mathbb{N}.$ Recently, Farhi proved a new identity: ${\rm lcm}({k \choose 0}, {k \choose 1}, >..., {k \choose k}) =\frac{{\rm lcm}(1, 2, ..., k+1)}{k+1}, \forall k\in \mathbb{N}.$ In this note, we show that Nair's and Farhi's identities are equivalent.