Solvability of $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)=\left(\genfrac{}{}{0ex}{}{2a}{a}\right)\left(\genfrac{}{}{0ex}{}{x+2b}{b}\right)$

Abstract

Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)=\left(\genfrac{}{}{0ex}{}{2a}{a}\right)\left(\genfrac{}{}{0ex}{}{x+2b}{b}\right)$ if and only if $x=a=1$. We do this by looking at different cases depending on the values of $x$ and $k$. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, Stănică, Shanta, Shorey and Nair are also used.