A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Singmaster’s Conjecture In The Interior Of Pascal’s Triangle
Tekijät: Matomäki Kaisa, Radziwiłł Maksym, Shao Xuancheng, Tao Terence, Teräväinen Joni
Kustantaja: OXFORD UNIV PRESS
Julkaisuvuosi: 2022
Journal: Quarterly Journal of Mathematics
Tietokannassa oleva lehden nimi: QUARTERLY JOURNAL OF MATHEMATICS
Lehden akronyymi: Q J MATH
Artikkelin numero: haac006
Sivujen määrä: 41
ISSN: 0033-5606
eISSN: 1464-3847
DOI: https://doi.org/10.1093/qmath/haac006
Rinnakkaistallenteen osoite: 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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