Related papers: Congruences for Wolstenholme primes

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…

The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ ($n=0,1,2,\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel…

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…

For a prime $p$ and an integer $a \in \Z$ we obtain nontrivial upper bounds on the number of solutions to the congruence $x^x \equiv a \pmod p$, $1 \le x \le p-1$. We use these estimates to estimate the number of solutions to the congruence…

Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem…

In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$ \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{(1)_k^3}4^k\equiv…

The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv…

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|=…

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Ap\'ery numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any…

In this paper we prove a new recurrence relation on the van der Waerden numbers, $w(r,k)$. In particular, if $p$ is a prime and $p\leq k$ then $w(r, k) > p \cdot \left(w\left(r - \left\lceil \frac{r}{p}\right\rceil, k\right) -1\right)$.…

In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv…

A computational proof is given for congruences modulo $p$ for the class equation $H_{-28p}(X)$, when the prime $p$ satisfies $p \equiv 3$ (mod $4$), and for the product $H_{-7p}(X) H_{-28p}(X)$, when $p \equiv 1$ (mod $4$).

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…

The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section,…