Singmaster's Conjecture In The Interior Of Pascal's Triangle

Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao*, Terence Tao, Joni Teräväinen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 ≥ 2, the number of solutions to the equation nm = t for natural numbers 1 ≤ m < n is bounded. In this paper we establish this result in the interior region log2/3+ ϵ n) ≤ m ≤ n - (log2/3+ ϵn) for any fixed > 0. Indeed, when t is sufficiently large depending on 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)(n-m+1) denotes the falling factorial.

Original languageEnglish (US)
Pages (from-to)1137-1177
Number of pages41
JournalQuarterly Journal of Mathematics
Volume73
Issue number3
DOIs
StatePublished - Sep 1 2022

ASJC Scopus subject areas

  • General Mathematics

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