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ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS

Part of: Sequences and sets Combinatorics Elementary number theory

Published online by Cambridge University Press:  08 March 2024

GUO-SHUAI MAO
GUO-SHUAI MAO*
Affiliation:
Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, PR China
*
e-mail: [email protected]
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Abstract

We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory 13(11) (2011), 2219–2238]. Let p be an odd prime. Then

$$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$

where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p-adic integers $a, b$.

Keywords

congruencescentral binomial coefficientsharmonic numbersBernoulli numbersEuler numbers

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 11A07: Congruences; primitive roots; residue systems 11B68: Bernoulli and Euler numbers and polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 409 - 420
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author was funded by the National Natural Science Foundation of China (grant nos. 12001288, 12071208).

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