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ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 08 March 2024, pp. 409-420
- Print publication:
- December 2024
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- Article
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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 
$a, b$.
