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

Matomäki, Kaisa and Radziwiłł, Maksym and Shao, Xuancheng and Tao, Terence and Teräväinen, Joni (2021) Singmaster's conjecture in the interior of Pascal's triangle. . (Unpublished)

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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 ε > 0. Indeed, when t is sufficiently large depending on ε, 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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Additional Information:Attribution 4.0 International (CC BY 4.0) KM was supported by Academy of Finland grant no. 285894. MR acknowledges the support of NSF grant DMS-1902063 and a Sloan Fellowship. XS was supported by NSF grant DMS-1802224. TT was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1764034. JT was supported by a Titchmarsh Fellowship.
Funding AgencyGrant Number
Academy of Finland285894
Alfred P. Sloan FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
James and Carol Collins ChairUNSPECIFIED
Mathematical Analysis and Application Research Fund EndowmentUNSPECIFIED
University of OxfordUNSPECIFIED
Record Number:CaltechAUTHORS:20210825-184547403
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110536
Deposited By: George Porter
Deposited On:25 Aug 2021 22:25
Last Modified:25 Aug 2021 22:25

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