Refereed journal article or data article (A1)

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




List of Authors: Matomäki Kaisa, Radziwiłł Maksym, Shao Xuancheng, Tao Terence, Teräväinen Joni

Publisher: OXFORD UNIV PRESS

Publication year: 2022

Journal: Quarterly Journal of Mathematics

Journal name in source: QUARTERLY JOURNAL OF MATHEMATICS

Journal acronym: Q J MATH

Number of pages: 41

ISSN: 0033-5606

eISSN: 1464-3847

DOI: http://dx.doi.org/10.1093/qmath/haac006

URL: https://doi.org/10.1093/qmath/haac006

Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/175192355


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 ((n)(m)) = t for natural numbers 1 <= m < n is bounded. In this paper we establish this result in the interior region exp(log(2/3+epsilon) n) <= m <= 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) . . . (n - m + 1) denotes the falling factorial.


Last updated on 2022-18-05 at 09:39