Related papers: Simpler congruences for Jacobi sum $J(1, 1)_{49}$ …

The aim of this paper is to deal with congruences for Jacobi sums of order $2l^{2}$ over a finite field $\mathbb{F}_{q}, q=p^{r}$, $p^{r}\equiv 1\ (mod \ 2l^{2})$, where $l>3$ and $p$ are primes. Further, we also calculate Jacobi sums…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain…

Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots…

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $\chi : \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic…

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}}…

In this work we consider the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1 \pmod n$ for each $k \in \mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ satisfies this…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize…

In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…

Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base 2. Recently, Z. H. Sun proved that \sum_{k=1}^{p-1}\frac{1}{k\cdot 2^k}\equiv q_p(2)-\frac{p}{2}q_p(2)^2 \pmod{p^2} which is a generalization of a…

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…

In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of…

It is shown that for any prime $p$ and any natural numbers $\ell, m,$ and $s$ such that $0<s<p$, the three following congruences \begin{align*}\sum_{i\ge \ell+1}(-1)^{m-i} {m \choose i}{m+s-1+i(p-1) \choose m+s-1+\ell(p-1)} &\equiv 0 \bmod…

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}}…

For $k=1,2,\ldots$ let $H_k$ denote the harmonic number $\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

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 $\alpha$ is $p$-integral, then we can completely determine $$…