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ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS
Published online by Cambridge University Press: 08 March 2024
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 
$a, b$.
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The author was funded by the National Natural Science Foundation of China (grant nos. 12001288, 12071208).
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