Necessary conditions for binomial collisions

Abstract

We shall give some necessary conditions for the equation $\left(\genfrac{}{}{0ex}{}{x}{a}\right)=\left(\genfrac{}{}{0ex}{}{y}{b}\right)$ to hold: if $\left(\genfrac{}{}{0ex}{}{2n+\delta }{n-m}\right)=\left(\genfrac{}{}{0ex}{}{2n+l}{n-k}\right)$ with $\delta =0$ or $1$, $0<m\le 0.735k$, $k<n$ and $n$ sufficiently large, then $l>(cn/\log n{)}^{40/21}$ for some constant $c$.

14 pages