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

TL;DR

This paper proves that in the interior of Pascal's triangle, each number occurs only a bounded number of times, specifically at most four solutions for large values, advancing understanding of Singmaster's conjecture in that region.

Contribution

The paper establishes bounds on the number of solutions to Pascal's triangle entries in the interior region, confirming Singmaster's conjecture in this specific area for large numbers.

Findings

At most four solutions for large t in the interior region

At most two solutions in each half of Pascal's triangle

Analogous bounds for falling factorial equations

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 $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, 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 $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial.