2 results
ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS
- Part of
-
- 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
-
- Article
-
- You have access
- HTML
- Export citation
-
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 
$a, b$.
Proof of some conjectural congruences involving Apéry and Apéry-like numbers
- Part of
-
- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 07 March 2024, pp. 508-527
-
- Article
-
- You have access
- HTML
- Export citation
-
In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then
\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}where
$A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.
