arXiv subjects
Publications and source records attributed to Guo-Shuai Mao.
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{6n+1}{(-512)^n}\binom{2n}n^3&\equiv p\left(\frac{-2}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}$ is the $n$-th Euler number.
In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\binom{2k}k\frac{3k+1}{(-16)^k}f_k\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ and $E_n$ stand for the $n$th Franel number and $n$th Euler number respectively.
In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv3\pmod 4$ be a prime. Then $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{8^k}\equiv-\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{(-16)^k}\pmod{p^3}.$$ And for any odd prime $p$, if $p=x^2+y^2$ with $4|x-1, 2|y$, then $$ \sum_{k=0}^{p-1}\frac{(k+1)\binom{2k}k^2}{8^k}+\sum_{k=0}^{(p-1)/2}\frac{(2k+1)\binom{2k}k^2}{(-16)^k}\equiv2\left(\frac{2}p\right)x\pmod{p^3}. $$
In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic number.
We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $α$ is $p$-integral, then we can completely determine $$ \sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k}\quad\text{and}\quad \sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binomα{k}\binom{-1-α}{k} $$ modulo $p^3$. In particular, by setting $α=-1/2$, we confirm two conjectured congruences of Z.-W. Sun.
In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.
In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, $$ and if $p\equiv2\pmod3$, then $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\frac{p-5}6}^{-2} \pmod{p^3}, $$ where $D_n=\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}$ stands for the $n$th Domb number.
In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number.
In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.
Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.
In this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in 2013. Let $p$ be an odd prime and let $a\in\mathbb{Z}^{+}$. Then if $p\equiv1\pmod3$, we have \begin{align*} \sum_{k=0}^{\lfloor\frac{5}6p^a\rfloor}\frac{\binom{2k}k}{16^k}\equiv\left(\frac{3}{p^a}\right)\pmod{p^2}, \end{align*} where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol.
In this paper, we prove a congruence which confirms a conjecture of Adamchuk. For any prime $p\equiv1\pmod3$ and $a\in\mathbb{Z}^{+}$, we have \begin{align*} \sum_{k=1}^{\frac{2}3(p^a-1)}\binom{2k}k\equiv0\pmod{p^2}. \end{align*}
In this paper, we generalise a congruence which proved by V. J.-W. Guo and J. Zeng \cite{gz-jnt-2012} involving Apéry numbers, and we obtain a congruence involving Franel numbers which confirms a congruence conjecture of Z.-W. Sun \cite[Conjecture 57(ii)]{sun-njdx-2019}.
In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}n^3\equiv p\left(\frac{-1}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\left(\frac14\right)\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}(x)$ are the Euler polynomials. This congruence confirms a conjecture of Sun \cite[(2.18)]{sun-numb-2019} with $n=1$.
In this paper, we prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv p+2\left(\frac{-1}{p}\right)p^3E_{p-3}\pmod{p^4},$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers.
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \end{align*} In fact, this supercongruence is a generalization of a supercongruence of van Hamme.
In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $δ\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/δ} \frac{\left(\frac12\right)^5_n}{n!^5}(10n^2+6n+1)(-4)^n &\equiv\begin{cases}p^{2r}\ \pmod{p^{r+4}} &\tt{if}\ r\leq4, \\0\ \pmod{p^{r+4}} &\tt{if}\ r \geq5. \end{cases} \end{align*}
Let $n$ and $r$ be positive integers. Define the numbers $S_n^{(r)}$ by $S_n^{(r)}=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}(2k+1)^r.$ In this paper we prove some conjectures of Guo and Liu which extend some conjectures of Z.-W. Sun \cite{Su1}, such as: There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that $$a_{2r-1}\sum_{k=0}^{n-1}S_k^{(2r-1)}\equiv0\pmod{n^2}\ \mbox{and}\ b_r\sum_{k=0}^{n-1}kS_k^{(r)}\equiv0\pmod{n^2}.$$ By Zeilberger algorithm, we find that for all $0\leq j<n$, $$(2j+1)\binom{2j}j\sum_{k=j}^{n-1}(2k-j+1)\binom kj^2\equiv0\pmod{n^2}.$$