On some congruence with application to exponential sums

Some Congruences of Kloosterman Sums and their Minimal Polynomials

We prove two results on Kloosterman sums over finite fields, using Stickelberger's theorem and the Gross-Koblitz formula. The first result concerns the minimal polynomial over Q of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27.

Congruences of alternating multiple harmonic sums

In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential.

Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case

There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.

Congruences for partial sums of the generating series for $\binom{3k}{k}$

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\binom{3k+e}{k}$. Our method derives such congruences directly from closed forms for the corresponding series.

Congruences of Power Sums

The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n, \end{cases}$ where $n\in{\mathbb N}$ and $p$ is prime. We extend this congruence, in particular, to the case when $p$ is any power of a prime. We also show that the sequence $(S_n(m) \text{ mod } k )_{m \geq 1}$ is periodic and determine its period.

Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case

We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and Müller from the continuous setting to various discrete settings, obtaining sharp bounds for exponential sums and the number of solutions to polynomial congruences for general quasi-homogeneous polynomials in two variables. This extends work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram.

Factors of certain sums involving central q-binomial coefficients

Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pan's technique and a simple $q$-congruence observed by Guo and Schlosser.

A Curious Congruence Involving Alternating Harmonic Sums

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 \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{{{(-1)}^{i}}}{ijk}}\equiv \frac{{{p}^{r-1}}}{2}{{B}_{p-3}}\, (\bmod \, {{p}^{r}}).\]

Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

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}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.

Congruences for sums of powers of an integer

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where $m(q,e)$ is "large". We also pay special attention to the situation where $e$ is a prime power.

New Congruences on Multiple Harmonic Sums and Bernoulli Numbers

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \ge 11$ and integer $r\ge 2$, we prove that $$ \sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{6}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{6}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\equiv - \frac{5!}{18}p^{r-1}B_{p-3}^{2} \pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernou…

A congruence modulo $n^3$ involving two consecutive sums of powers and its applications

For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\ge 2 $$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$$ The above congruence allows us to state an equivalent formulati…

Small Solutions of Quadratic Congruences, and Character Sums with Binary Quadratic Forms

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …

Some 3-adic congruences for binomial sums

We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.

Congruences of multiple sums involving invariant sequences under binomial transform

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd} -{2^{n+1}+4\over n6^n} B_{p-n}({1\over 3}) &\pmod{p} &{if $n$ is even}. $$ where $n$ is a positive integer and $p$ is prime such that $p>\max(n+1,3)$.

Congruences on sums of $q$-binomial coefficients

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.

Congruences on sums of super Catalan numbers

In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.

不同素数乘积模下的整数幂和的同余关系

A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes

本文证明了在特定条件下，不同素数乘积模下的整数幂和满足一种同余关系。

A New Super Congruence Involving Multiple Harmonic Sums

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{5}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{5}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\equiv -\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \pmod{{{p}^{r}}}. \end{equation} This gives an extension of a family of super congruences found by Wang, Cai and Zhao.

关于涉及$\binom{rk}{k}$的和的同余式

Congruences for sums involving $\binom{rk}{k}$

研究了模$p$下形式为$\sum_k\binom{rk}{k}x^k/k$的有限和的同余性质，通过两种互补方法获得模$p^2$下的同余式，用有限多项式对数表达式表示。

互反幂和与广义莱默型乘积的高阶同余

Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

研究互反幂和与广义莱默型乘积的高阶同余，将互反平方和结果扩展到奇数次互反和并建立模\(n\)统一同余，通过完全指数贝尔多项式解决广义莱默型乘积高阶同余无简单闭式问题，提供相关显式计算与验证框架。

Congruences for certain lacunary sums of products of binomial coefficients

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 p\\ \sum_{i\ge 0}(-1)^{m-i} {m \choose i}{\ell+ip \choose m+s-1}&\equiv 0 \bmod p^m\\ \sum_{j,i\ge \ell}(-1)^{j-i}{m \choose j} {j \choose i}{j+s-1+i(p-1) \choose j+s-1+\ell(p-1)}&\equiv 0 \bmod p^{m-\ell} \end{align*} hold true. The corresponding quotients involve Adelberg polynomials which can be computed explicitly, providing closed-form expressions for t…

关于幂和与Kummer同余

On Power Sums and the Kummer Congruences

本文通过研究多重集在乘法作用下的稳定化子，揭示了元素层面的底层结构，为经典Kummer同余提供了新证明，并提出了基于局部信息组合的新视角。

Some $q$-analogs of congruences for central binomial sums

We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.

多重指数和及其在二次同余式中的应用

Multiple exponential sums and their applications to quadratic congruences

本文提出一种评估多变量一般指数和的方法，并应用于二次同余式，得到其解数的渐近公式及均匀分布结果。

On solving a restricted linear congruence using generalized Ramanujan sums

Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define $\mathcal{C}_{j,s} = \{1\leq x\leq n^s | (x,n^s)_s = d^s_j\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\s…

Double exponential sums and congruences with intervals and exponential functions modulo a prime

Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd (a,p)=1, $$ where $\mathcal{N}$ and $\mathcal{M}$ are intervals of consecutive integers with $|\mathcal{N}|=N$ and $|\mathcal{M}|=M<T$ elements. One representative example is the following consequence of the main result: if $N=M\approx p^{1/3}$, then $|S|< N^{2-1/8 + o(1)}$. We then apply our estimate to obtain new results on additive congruences involving …

素数和与伯努利多项式的同余性质

Congruence properties of prime sums and Bernoulli polynomials

本文研究涉及素数的特定和式的同余性质，利用伯努利数与伯努利多项式表达结果，并进一步导出一个一般的同余关系。

P-递归序列部分和的同余式的简化

Reduction on the congruences of partial sums of P-recursive sequences

本文简化了P-递归序列部分和同余式的证明方法，放宽望远镜算子要求，利用序列同余性质，确认了孙的猜想并发现中心三项式系数的新同余式。

Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…

On supercongruences for truncated sums of squares of basic hypergeometric series

Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.

Multiple harmonic sums and Wolstenholme's theorem

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mod{p^{2n+3}}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose members contain a variable number of terms, and show that in this family …

Kloosterman sums with primes and solvability of one congruence with inverse residues. I

In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus $q$. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q} \] in prime variables $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-δ}$. Here $g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, where $a,b,m$ and $k\ge 3$ are arbitrary integers, $(ab,q)=1$. The main result of the paper gives an asymptotic formula for the number of solutions for the case when $q$ is coprime …

Sums of products of congruence classes and of arithmetic progressions

Consider the congruence class R_m(a)={a+im:i\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b)+R_m(c)R_m(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+\ell m) for some i,j,k,\ell\in Z. It is proved that if gcd(a,b,c,d,m)=1, then R_m(a)R_m(b)+R_m(c)R_m(d) is equal to the congruence class R_m(ab+cd), and that the sum of products set P_m(a)P_m(b)+P_m(c)P_m(d) eventually coincides with the infinite arithmetic progression P_m(ab+cd).

涉及多重调和和的一些新奇特同余式

Some new curious congruences involving multiple harmonic sums

本文通过应用多重调和和的同余式，推广了已知的奇特同余式，得到了多个新的一般化结果，并给出了具体示例。

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

Telescoping method and congruences for double sums

In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.

关于正则分拆函数、广义 Tau 函数与分拆函数加权复合和的同余关系

Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums

本文将广义 τ_k(n) 表示为二项式系数加权分拆和，从而建立其与 t-正则分拆函数 R_t(n) 及加权复合和之间的同余恒等式。

Asymptotic relations for weighted finite multiple zeta values

Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike res…

Congruences with intervals and arbitrary sets

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results…

Congruences related to Eulerian and Euler numbers

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

Proofs of Some Conjectures of Chan on Appell-Lerch Sums

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$ϕ(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=ϕ(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured …

Exponential Sums and Congruences with Factorials

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials $n!m!$ with $\max\{n,m\}<p^{1/2+ε}$ are uniformly distributed modulo $p$, and that any residue class modulo $p$ is representable in the form $m!n!+n_1! + ... +n_{49}!$ with $\max \{m,n, n_1, >..., n_{49}\} < p^{8775/8794+ ε}$.

On some congruences using multiple harmonic sums of length three and four

In the present paper, we determine the sums $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}}{j^{s_2}}$ and $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}H_j^{(s_4)}}{j^{s_2}}$ modulo $p$ and modulo $p^2$ in certain cases. This is done by using multiple harmonic sums of length three and four, as well as, many other results. In addition, We recover three congruences conjectured by Z.-W Sun and solved later by the author himself and R. Meštrović.

Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the…

A congruence involving harmonic sums modulo $p^αq^β$

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $ Z(n)=\sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{1}{ijk}$ and $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $α,~β$, \begin{equation*} Z(p^αq^β)\equiv 2(2-q)(1-\frac{1}{q^{3}})p^{α-1}q^{β-1}B_{p-3}\pmod{p^α} \end{equation*} and the necessary and sufficient co…

Combinatorial Sums $\sum_{k\equiv r(\mbox{mod} m)}{n\choose k}a^k$ and Lucas Quotients (II)

In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed in terms of some recurrent sequences with orders not exceeding $φ{(m)}$ and give some new congruences.

A note on the Voronoi congruences and the residue of the Fermat quotient

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 divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…

Congruences involving the reciprocals of central binomial coefficients

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

与富克斯群相关的广义戴德金和的变换法则

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

该研究建立了与非全纯艾森斯坦级数克罗内克极限函数及高阶变体相关的广义戴德金和变换法则，适用于第一类富克斯群，并给出多类群的示例。

$N$-tuple sum analogues for Ramanujan-type congruences

In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.

Additional congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^{2}\geq1$ equal to squared integers $s^{2}$, has no solutions if $M\equiv3,5,6,7,8$ or $10\left(mod\,12\right)$ and has integer solutions if $M\equiv0,9,24% or %33\left(mod\,72\right)$; or $M\equiv1,2$ or $16\left(mod\,24\right)$; or $M\equiv11\left(mod\,12\right)$. In this paper, additional congruence conditions are demonstrated on the allowed values of $M$ that yield solutions to the problem by using Beeckmans' eight necessary conditions, refining further the possibl…

Linear Congruences with Ratios

We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences $$ \sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p, $$ with variables from rather general sets.