Products of Factorials Modulo p

On the parity of the multiplicative order of certain products of integers related to Gauss factorials

In this note, we prove that under some conditions, certain products of integers related to Gauss factorials are always quadratic residues.

Character Sums and Congruences with n!

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+ε}, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a \equiv n_1! ... n_7! \pmod p, where $\max \{n_i | i=1,... 7\}=O(p^{11/12+ε}), and we find the asymptotic formula for …

因子阶乘的乘积 which 是阶乘的乘积

Products of factorials which are products of factorials

本文在显式abc猜想下证明了因子阶乘乘积方程在特定子集中有有限多个非平凡解。

On asymptotic constants related to products of Bernoulli numbers and factorials

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., \[ \prod_{ν=1}^n |B_{2ν}| \quad \text{and} \quad \prod_{ν=1}^n (k ν)!^{ν^r} \quad \text{as} \quad n \to \infty \] for integers $k \geq 1$ and $r \geq 0$. Our main interest is to determine exact expressions, in terms of known constants, for the asymptotic constants of these expansions and to show some relations among them.

On members of Lucas sequences which are products of factorials

Here, we show that if $\{U_n\}_{n\ge 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=m_1!m_2!\cdots m_k!$ with $1<m_1\le m_2\le \cdots\le m_k$ satisfies $n<3\times 10^5$. We also give better bounds in case the roots of the Lucas sequence are real.

Erdos问题的解：阶乘的不同剩余

Solution to the Erdos problem on distinct residues of factorials

本文用初等方法证明，对于大于5的素数p，2!, 3!, ..., (p-1)!模p的剩余不可能全部不同。

On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad D_{m_i}\in \{B_{m_i}, C_{m_i}\}$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12, 16\}$. Here $B_m$ is the middle binomial coefficient $\binom{2m}{m}$ and $C_m$ is the Catalan number $\frac{1}{m+1}\binom{2m}{m}$.

固定幂带中阶乘素数赋值的加权等分布

Weighted equidistribution of factorial prime valuations in fixed power bands

研究固定幂带内素数在阶乘中赋值的对数质量分布，建立渐近公式并推导多项式-阶乘方程解的必要容量条件。

在素数模下奇异的自守乘积

Automorphic products that are singular modulo primes

本文利用O(n, 2)上的Rankin--Cohen括号，证明了反射Borcherds乘积的傅里叶系数通常满足模某些素数的同余关系。

广义黎曼假设下算术级数中欧拉阶乘级数的显式结果

Explicit results for Euler's factorial series in arithmetic progressions under GRH

在广义黎曼假设下，研究欧拉阶乘级数在p-adic域中算术级数上的线性形式非零性，给出显式下界及素数分布区间。

关于序列$n! \bmod p$的研究

On the sequence $n! \bmod p$

该研究证明阶乘序列模素数p的不同剩余类下界，给出区间内阶乘剩余类的相关结论，并推导出每个非零剩余类可表为七个阶乘的模p乘积，较先前结果有多项式级改进。

On Ecalle's and Brown's polar solutions to the double shuffle equations modulo products

Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.

Tribonacci数之积为阶乘之积

Products of Tribonacci Numbers that are the Products of Factorials

本文推广了Tribonacci数与阶乘乘积方程的解，给出了连续Tribonacci项乘积等于阶乘乘积的所有解。

On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(πx_k)$ for $N \to \infty$, where the numbers $ω=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $ω$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $ω$ are the initial segment of a Kronecker or van der Corput sequence. The paper concludes with some probabilistic a…

阶乘的平方乘积及 Erdős 与 Graham 的一个猜想

Square products of factorials and a conjecture of Erdős and Graham

本文研究 Erdős 与 Graham 关于阶乘乘积为完全平方数的计数函数 $D_k(X)$ 的增长阶猜想，证明 $D_3(X)$ 有显式渐近公式，$D_5$ 与 $D_6$ 均为 $X$ 量级，并完全确定所有 $k$ 的增长阶。

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。

区分模p的椭圆曲线并识别乘积表示的像

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

该研究针对定义在ℚ上的两条椭圆曲线与有理素数p，借助古尔萨引理刻画其剩余伽罗瓦表示乘积的所有可能像，定义见证比率并阐明其计算效用及与斯特姆界的关联。

Polynomial products modulo primes and applications

For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ We apply this to improve the lower bound on the number of distinct quadratic fields of the form $\mathbb{Q}(\sqrt{F_P(n)})$ in short intervals $M\le n\le M+H$ previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of $F_P(n)(\text{mod}\ p)$ for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and oth…

A Generalization of A Result of Gauss on Primitive Root

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.

关于Ecalle和Brown解决双洗关系模乘积的注记

Notes on Ecalle's and Brown's solutions to the double shuffle relations modulo products

本文探讨了Brown和Ecalle提出的双洗关系模乘积下的极/多项式解之间的联系。

The Massey Vanishing Conjecture for fourfold Massey products modulo 2

We prove the Massey Vanishing Conjecture for $n=4$ and $p=2$. That is, we show that for all fields $F$, if a fourfold Massey product modulo $2$ is defined over $F$, then it vanishes over $F$.

整数与其模逆乘积的均值

Mean values of the product of an integer and its modular inverse

本文将 Chan (2016) 关于素数模下整数与其乘法逆元乘积均值的最新结果，推广至复合数模的情形。

Gamma、伪Gamma与逆Gamma函数的级数与乘积表示

Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions

本文利用牛顿插值级数研究Gamma函数，给出其乘积与级数表示，并由此得到倒数Gamma函数泰勒系数的新公式及Euler-Mascheroni常数仅含有理项的两种级数表示，同时引入在整个实轴收敛的伪Gamma函数$Λ$，并猜想逆Gamma函数主支的新级数表示。

Congruences with intervals and subgroups modulo a prime

We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),

Products of primes in arithmetic progressions: a footnote in parity breaking

We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$.

Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\equiv λ\pmod p; \quad x\in \N, \quad L<x<L+p/n, $$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n, λ$ and $L$, for some absolute constant $c>0$. This implies, in part…

On the q-binomial identities involving the Legendre symbol modulo 3

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

Quadratic residues and quartic residues modulo primes

In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{cases}-(A^2+4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=1, \\(-A^2-4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2+5ij+2j^2…

Breaking the 6/5 threshold for sums and products modulo a prime

Let $A \subset \mathbb{F}_p$ of size at most $p^{3/5}$. We show $$|A+A| + |AA| \gtrsim |A|^{6/5 + c},$$ for $c = 4/305$. Our main tools are the cartesian product point--line incidence theorem of Stevens and de Zeeuw and the theory of higher energies developed by the second author.

The Catalan numbers have no forbidden residue modulo primes

Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the product of central binomial coefficients modulo $p$. The directed sub-graph of the automata for $C_n \mod p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.

高阶Yoneda积结构与模$p$的Iwasawa代数

Higher Yoneda product structures and Iwasawa algebras modulo $p$

本文回答了Sorensen提出的三个问题，研究了无挠pro-$p$群的模$p$ Iwasawa代数与其Yoneda代数上的$A_\infty$-代数结构之间的关系。

一个反例驳斥萨尔科齐关于素数模下和积猜想

A counterexample to a conjecture of Sárközy on sums and products modulo a prime

本文构造了反例，证明萨尔科齐关于素数模下和积猜想不成立，指出当集合大小为(p-1)/2时，1不在和积集中，而当大小超过p/2时，和集等于素数域乘法群。

Factorial ratios, hypergeometric series, and a family of step functions

We give a complete classification of a certain family of step functions related to the Nyman--Beurling approach to the Riemann hypothesis and previously studied by V. I. Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations are made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nF_{n-1}. We also note applications to the classification of cyclic quotient singularities.

Product of three primes in large arithmetic progressions

For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Final…

Products of quadratic residues and related identities

In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.

Products in Residue Classes

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.

2-adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman--Friedland--Loewy on the parity of the degrees of projective varieties of $n\times n$ complex symmetric matrices of rank at most $k$. We also characterize the parity of the degrees of projective varieties of $n\times n$ complex skew symmetric matrices of rank at most $2p$. We give recursive relations which determine the parity of the degrees of projective varieties of $m\times n$ complex matrices of rank at most $k$. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of $n\times n$ skew symmetric real matr…

涉及两个二项式系数乘积的超同余式模 $p^4$

Supercongruences involving products of two binomial coefficients modulo $p^4$

本文证明了关于两个二项式系数乘积的超同余式模p⁴的猜想，涉及调和数和二项式系数的求和问题。

Size of product of a number and its multiplicative inverse, Moments of L-functions and Exponential Sums

In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple exponential sum.

Linear complexity and trace representation of quaternary sequences over $\mathbb{Z}_4$ based on generalized cyclotomic classes modulo $pq$

We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences.

Geometric configurations in the ring of integers modulo $p^{\ell}$

We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.

Integrality of hook ratios

We study integral ratios of hook products of quotient partitions. This question is motivated by an analogous question in number theory concerning integral factorial ratios. We prove an analogue of a theorem of Landau that already applied in the factorial case. Under the additional condition that the ratio has one more factor on the denominator than the numerator, we provide a complete classification. Ultimately this relies on Kneser's theorem in additive combinatorics.

Residues modulo powers of two in the Young-Fibonacci lattice

We study the subgraph of the Young-Fibonacci graph induced by elements with odd $f$-statistic (the $f$-statistic of an element $w$ of a differential graded poset is the number of saturated chains from the minimal element of the poset to $w$). We show that this subgraph is a binary tree. Moreover, the odd residues of the $f$-statistics in a row of this tree equidistibute modulo any power two. This is equivalent to a purely number theoretic result about the equidistribution of residues modulo powers of two among the products of distinct odd numbers less than a fixed number.

Proofs of Modulo 11 and 13 Cylindric Kanade-Russell Conjectures for $A_2$ Rogers-Ramanujan Type Identities

We present proofs of two new families of sum-product identities arising from the cylindric partitions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured by Kanade-Russell in the spirit of Andrews-Schilling-Warnaar identities of the $A_2$ Rogers-Ramanujan type. We follow the footsteps of Kanade-Russell while we alter the computations heavily to accomplish our goals.

Congruences for sums involving products of three binomial coefficients

Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3},\frac 1{2k-1},\frac 1{k+2}, \frac 1{k+3}, k,k^2,k^3,\frac 1{a+k},\frac 1{a+k-1}.$$ As consequences, taking $a=-\frac 12,-\frac 13,-\frac 14,-\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.

Remarks on the Gaudin model modulo $p$

We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In …