Yossi Moshe
On a Problem of Erd\H os regarding Binomial Coefficients
The paper is published: Discrete Math. 297 (2005) 91-103

MSC:

11B65 Binomial coefficients; factorials; $q$-identities, See also {05A10, 05A30}

11D99 None of the above but in this section

11A99 None of the above but in this section

Abstract: Let $d$ be a positive rational number.
We prove the existence of infinitely
many
pairs $(A,B)$ of disjoint finite subsets of $\mathbb{N}$ with
$\prod\limits_{a\in A}\!\binom{2a}{a}~\!\!=~\!\!d\!\prod\limits_{b\in B}\!\binom{2b}{b}$.
This solves in particular a problem of Erd\H{o}s~\cite{ER}.

Keywords: Binomial coefficients, middle binomial coefficients, diophantine equations