Singmaster's conjecture in the interior of Pascal's triangle

Abstract

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t\ge 2$, the number of solutions to the equation $\left(\genfrac{}{}{0ex}{}{n}{m}\right)=t$ for natural numbers $1\le m<n$ is bounded. In this paper we establish this result in the interior region $\exp ({\log }^{2/3+\epsilon }n)\le m\le n-\exp ({\log }^{2/3+\epsilon }n)$ for any fixed $\epsilon >0$. Indeed, when $t$ is sufficiently large depending on $\epsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n{)}_m=t$, where $(n{)}_m:=n(n-1)\dots (n-m+1)$ denotes the falling factorial.

33 pages