Abstract
Abstract
We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory13(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
$a, b$
.
Publisher
Cambridge University Press (CUP)
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