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Published September 2022 | Submitted
Journal Article Open

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


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+ε)n) ≤ m ≤ n − exp(log^(2/3+ε)n) for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, 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.

Additional Information

© The Author(s) 2022. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Received: 06 August 2021; Accepted: 08 March 2022; Corrected and typeset: 05 April 2022; Published: 05 April 2022. We thank William Verrault for drawing attention to the recent preprint [11], Antoine Deleforge for pointing out a typo in the original version of this manuscript, and Jordan Ellenberg for noting a connection between Lemma 2.2 and the Bombieri–Pila determinant method. K.M. was supported by Academy of Finland grant no. 285 894. M.R. acknowledges the support of National Science Foundation grant DMS-1 902 063 and a Sloan Fellowship. X.S. was supported by National Science Foundation grant DMS-1 802 224. T.T. was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by National Science Foundation grant DMS-1 764 034. J.T. was supported by a Titchmarsh Fellowship.

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August 20, 2023
October 23, 2023