A1 Refereed original research article in a scientific journal
Singmaster’s Conjecture In The Interior Of Pascal’s Triangle
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
Article number: haac006
Number of pages: 41
ISSN: 0033-5606
eISSN: 1464-3847
DOI: https://doi.org/10.1093/qmath/haac006
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/175192355
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.
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