A q-analogue of Lehmer's congruence

Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings

In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.

A q-analogue of Wolstenholme's harmonic series congruence

We establish a q-analogue of Wolstenholme's harmonic series congruence.

Notes on q-analogues of a binomial congruence of Glaisher

Andrews once gave $q$-analogues of a binomial congruence of Glaisher, and he suggested perfect $q$-analogues. In this note we give ones meeting the demand of Andrews.

An inductive proof of Straub's q-analogue of Ljunggren's congruence

Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.

关于模素数广义Lehmer数存在性的研究

On existence of generalised Lehmer numbers modulo primes

本文推广Lehmer数为模奇素数$q$的$q$-元组，计数并给出带显式误差项的渐近估计，利用特征和与Deligne界。

Analogues of Lehmer's conjecture in positive characteristic

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let $ϕ:A\to\text{End}(\mathbb{G}_a)$ be a Drinfeld $A$-module of rank $r$ defined over a finite extension $L$ of $K$ and $\hat{h}_ϕ$ its canonical height. Given a non-torsion point $α$ of $ϕ$ of degree $d$ over $K$, we prove that $\hat{h}_ϕ(α)\ge 1/d$. A similar statement is proved for the canonical height of a point of infinite order of a non-constant semi-stable elliptic curve defined over…

A $q$-analogue of Wilson's congruence

Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{σ\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,σ} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\mathfrak f}_{n-1}(q)\equivμ(n)\pmod{Φ_n(q)}, $$ where $μ$ denotes the Möbius function and $Φ_n(q)$ is the $n$-th cyclotomic polynomial.

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.

关于林的分区函数类比的同余式

Congruences for an analogue of Lin's partition function

研究了林受限分区函数类比B(n)的算术性质，通过q级数技术和模函数建立模2、3、5、7和9的拉马努金型同余式。

Operator analogues of Mahler's measure

Motivated by a geometric meaning of Mahler's measure, we introduce two operator analogues of Mahler's measure. This leads to some interesting equalities and inequalities between the two operator-theoretic Mahler measures and the classical Mahler measure. In order to apply these results to the operator version of Lehmer's problem, we introduce and study an important class of operators, the so-called subharmonic operators. It is shown that the operator version of Lehmer's problem fails under some mild condition.

Polynomial analogues of Ramanujan congruences for Han's hooklength formula

This article considers the eta power $\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.

Some q-congruences related to 3-adic valuations

In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore $$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of the former congruence was conjectured by Guo and Zeng. In this paper we prove the Guo-Zeng conjecture and also give a $q$-analogue of the latter congruence.

On a congruence involving $q$-Catalan numbers

Based on a $q$-congruence of the author and Petrov, we set up a $q$-analogue of Sun--Tauraso's congruence for sums of Catalan numbers, which extends a $q$-congruence due to Tauraso.

Van Hamme 若干同余式的一个精细 q-模拟

A refined $q$-analogue of some congruences of Van Hamme

本文受 Guo 近期工作启发，为 Van Hamme 的 (B.2)、(E.2)、(F.2) 超同余式建立了统一的 q-模拟，并由此导出与 Euler 多项式相关的含参数超同余式。

A congruence involving products of $q$-binomial coefficients

In this paper we establish a $q$-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.

Congruences arising from Apéry-type series for zeta values

Recently, R. Tauraso established finite $p$-analogues of famous Apéry series for $ζ(2)$ and $ζ(3).$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Apéry-type series for $ζ(4)$ and $ζ(5).$ We also prove a $p$-analogue of Zeilberger's series for $ζ(2)$ confirming a conjecture of Z. W. Sun.

$N$-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.

Some congruences related to the q-Fermat quotients

We give q-analogues of the following congruences by Z.-W. Sun: \sum_{k=1}^{p-1}\frac{D_k}{k} \equiv -\frac{2^{p-1}-1}{p} \pmod p,\\ \sum_{k=1}^{p-1}\frac{H_k}{k 2^k}\equiv 0 \pmod{p},\quad p\geqslant 5, where p is a prime, D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k} are the Delannoy numbers, and H_n=\sum_{k=1}^n\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \sum_{1\leqslant k_1\leqslant \cdots \leqslant k_m\leqslant p-1}\frac{1}{k_1\cdots k_m 2^{k_m}} \equiv\frac{1}{2}\sum_{k=1}^{p-1}\frac{(-1)^{k-1}}{k^m} \pmod p, which is a multiple g…

Van Hamme的G.2超同余的q-模拟

q-analogues of the G.2 Supercongruence of Van Hamme

该研究受同余与q-同余研究启发，用创造性显微术方法首次给出Van Hamme的（G.2）超同余的两种直接q-模拟，还提出了与Swisher的Dwork型超同余（G.3）相关的猜想。

Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas

For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\frac{1}{64^k} \equiv 0 \pmod{p^2} \quad\text{for}\quad p\equiv 3\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^…

Congruences for $q$-binomial coefficients

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial ratios.

q-Analogues of two "divergent" Ramanujan-type supercongruences

Guillera and Zudilin proved three "divergent" Ramanujan-type supercongruences by means of the Wilf-Zeilberger algorithmic technique. In this paper, we prove $q$-analogues of two of them via the $q$-WZ method. Additionally, we give $q$-analogues of two related congruence of Sun, one is confirmed and the other is conjectural.

New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series

In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite…

A common $q$-analogue of two supercongruences

We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad \sum_{k=0}^{(p-1)/2}A_k\equiv a(p)\pmod{p^2}, $$ where $p>2$ is prime, $$ A_k=\prod_{j=0}^{k-1}\biggl(\frac{1/2+j}{1+j}\biggr)^3=\frac1{2^{6k}}{\binom{2k}k}^3 \quad\text{for}\ k=0,1,2,\dots, $$ and $a(p)$ is the $p$-th coefficient of (the weight 3 modular form) $q\prod_{j=1}^\infty(1-q^{4j})^6$. We complement our result with a general common $q$-congruence for related …

$SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ 函数及其类似物的一些新同余式和恒等式

Some new congruences and identities for $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ functions and analogues

本文证明了 $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ 函数的新恒等式和同余式（包括单调性），并推广到一般分拆族 $S_{\mathcal P}(n)$ 和彩色分拆类似物，证明了可除性性质。

Congruence preserving functions in the residue class rings of polynomials over finite fields

In this paper, as an analogue of the integer case, we define congruence preserving functions over the residue class rings of polynomials over finite fields. We establish a counting formula for such congruence preserving functions, determine a necessary and sufficient condition under which all congruence preserving functions are also polynomial functions, and characterize such functions.

Guo的两个$q$-同余猜想的证明

Proofs of two $q$-congruence conjectures of Guo

本文证明了Guo提出的两个q-同余猜想，分别涉及截断q-模拟的平方分圆同余和带参数s的截断基本超几何和的可除性。

广义Frobenius分拆的算术性质

Arithmetic properties of generalized Frobenius partitions

本文研究了广义m色Frobenius分拆函数cϕ_m在m为5、7、11时的同余性质，结合模形式理论与计算证明了仅有六个类似Ramanujan的同余式存在。

关于四项式系数的同余

Congruences concerning quadrinomial coefficients

本文建立了涉及四项式系数的模 $p^2$ 同余式，作为 Elkhiri 和 Mihoubi 关于三项式系数同余式结果的类比。

Agoh's conjecture: its generalizations, its analogues

In this paper we formulate two generalizations of Agoh's conjecture. We also formulate conjectures involving congruence modulo primes about hyperbolic secant, hyperbolic tangent, Nörlund numbers, as well as about coefficients of expansions in powers of other analytic functions. We formulate a thesis about combinatorial objects that do not produce fake primes.

由截断Appell级数产生的超同余式

Supercongruences Arising from Truncated Appell Series

研究截断Appell级数和截断q - Appell级数的同余性质，利用组合恒等式和‘创造性显微镜’方法，给出截断Appell级数\(F_1\)和\(F_2\)一些同余式的q - 类似，建立与\(F_4\)相关新q - 超同余式并猜想。

On $q$-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to $q$-binomial identities and prove $q$-analogues of two-one formulas…

行列式曲面上整数点的分布及其模p类比

Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue

本文建立了形如 $xy-zw=r$ 的方程（$r$ 为非零整数）的整数解带光滑权重的计数渐近公式，给出显式主项和关于变量 $x,y,z,w$ 及 $r$ 的强误差界；同时建立了同余式 $xy-zw \equiv 1 \pmod{p}$（$p$ 为大素数）的整数解带光滑权重的计数渐近公式，并给出强误差界。

矩形高斯二项式系数的Ljunggren--Jacobsthal型与Bailey型同余式

Ljunggren--Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients

本文针对模4余3的有理素数，研究矩形高斯二项式系数，证明其p-整性，建立Ljunggren--Jacobsthal型超同余式以解决Kalinin猜想的惰性素数情形，还给出矩形Bailey型模p同余式并完成相关证明。

函数域素数竞赛中的平局

Ties in Function Field Prime Races

本文研究函数域中素数分布的平局现象，通过两种方法证明了在特定条件下存在无限多个平局情况，包括特征2的情形。