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

Abstract

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

3 pages