On a conjectural congruence of Guo

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.

Proof of some conjectural congruences involving Domb numbers

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, $$ and if $p\equiv2\pmod3$, then $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\frac{p-5}6}^{-2} \pmod{p^3}, $$ where $D_n=\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}$ stands for the $n$th Domb number.

Gross 精细类数公式的证明

A proof of the refined class number formula of Gross

本文针对 Gross 于1988年提出的、联系 Stickelberger 元与代数调节子的精细类数公式猜想，给出了该同余式的证明。

来自Karlsson-Minton求和公式的一些同余式

Some congruences from the Karlsson-Minton summation formula

本文利用Karlsson-Minton求和公式证明了两个超同余式，作为Deines等人超同余式的变体，证实了V.J.W. Guo的猜想。

Non-Negative Integer Linear Congruences

We consider the problem of describing all non-negative integer solutions to a linear congruence in many variables. This question may be reduced to solving the congruence $x_1 + 2x_2 + 3x_3 + ... + (n-1)x_{n-1} \equiv 0 \pmod n$ where values of the unknowns, $x_i$, are sought among the non-negative integers. We consider the monoid of solutions of this equation and prove a conjecture of Elashvili concerning the structure of these solutions. This yields a simple algorithm for generating most (conjecturally all) of the high degree indecomposable solutions of the equation.

Detecting large simple rational Hecke modules for $Γ_0(N)$ via congruences

We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(Γ_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log…

Some $q$-congruences arising from certain identities

In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{Φ_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{Φ_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $Φ_…

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mor…

Nadji、Ahmia 和 Ramírez 关于双正则超分拆同余式的猜想

A conjecture of Nadji, Ahmia and Ram\'ırez on congruences for biregular overpartitions

本文通过构造精确生成函数，证明了双正则超分拆数 $\overline{B}_{4,3}(n)$ 在特定算术级数上模 $2$ 高次幂的同余式，推广了近期猜想，并进一步提出无限族线性同余猜想。

Some parametric congruences involving generalized central trinomial coefficients

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}}{12^k}T_k\equiv\left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{T_kH_k}{3^k}\equiv\frac{3+\left(\fr…

Symbolic summation methods and hypergeometric supercongruences

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 p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.

素数基二项式同余的结构化解

Structured Solutions of Prime-Base Binomial Congruences

研究素数基 q 下二项式同余式 C(qn, n) ≡ q^n (mod n) 的解结构，将解 n = q^t p 转化为模 p 同余和数字和不等式两个独立条件，并用于生成新解及证明平方解与 Wieferich 素数等价。

On a congruence involving harmonic series and Bernoulli numbers

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\ı+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ where $B_n$ is the $n$-th Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime $p$ in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer $n$ with an odd prime fac…

New series for powers of $π$ and related congruences

Via symbolic computation we deduce 97 new type series for powers of $π$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k \binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} =\frac{18\times557403^3\sqrt{10005}}{5π}$$ with \begin{align*}P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $$\sum_{k=1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3\binom{2k}k^3} = \frac{π^2-8}2,$$ and $$\sum_{n=0}^\infty\frac{3n+1}{(-100)^n} \sum_{k=0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8π},$$ …

Parity of 3-regular partition numbers and Diophantine equations

Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime …

A parametric congruence motivated by Orr's identity

For any $m,n\in\mathbb{N}=\{0,1,2\ldots\}$, the truncated hypergeometric series ${}_{m+1}F_m$ is defined by $$ {}_{m+1}F_m\bigg[\begin{matrix}x_0&x_1&\ldots&x_m\\ &y_1&\ldots&y_m\end{matrix}\bigg|z\bigg]_n=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_m)_k}{(y_1)_k\cdots(y_m)_k}\cdot\frac{z^k}{k!}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol. Let $p$ be an odd prime. For $α,z\in\mathbb{Z}_p$ with $\langle -α\rangle_p\equiv0\pmod{2}$, where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$ for any $x\in\mathbb{Z}_p$, we mainly prove the following congruen…

Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k} \end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k…

Proof of some supercongruences concerning truncated hypergeometric series

In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$ \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{(1)_k^3}4^k\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}} $$ and $$ \sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac12)_k^4}{(1)_k^4}\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol and $B_0,B_1,B_2,\ldots$ are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. App…

Divisibility results concerning truncated hypergeometric series

In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $$\sum_{k=0}^{p-1}\frac{(q-\frac{p}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3} $$ and $$p^n\sum_{k=0}^{p-1}\frac{(1)_k^n}{(\frac{p}{n}-q+2)_k^n}\equiv0\pmod{p^3}$$ for any prime $p>\max\{n,(q-1)n+1\}$, where $(x)_k$ denotes the Pochhammer symbol defined by $$ (x)_k=\begin{cases}1,\quad &k=0,\\ x(x+1)\cdots(x+k-1),\quad &k>0.\end{cases}$$ Let $n\geq4$ be an even integer. Then for a…

On sums involving products of three binomial coefficients

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+d}}{64^k}\equiv 0\pmod{p^2}$$ for all $d\in\{0,\ldots,p-1\}$ with $d\equiv (p+1)/2\pmod2$. If $p\equiv 1\pmod4$ and $p=x^2+y^2$ with $x\equiv 1\pmod4$ and $y\equiv 0\pmod2$, then we show $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+1}}{(-8)^k}\equiv 2p-2x^2\pmod{p^2}\ \ \mbox{and}\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{2k}{k+1}^2}{(-8)^k}\equiv-2p\pmod{p^2}…

Artin-Hasse 级数模素数的某些系数

On some coefficients of the Artin-Hasse series modulo a prime

本文研究 Artin-Hasse 指数模奇素数 $p$ 的系数，给出偶数指标系数的多项式递推，证明一个函数方程的模 $p$ 类似物，并由此推导涉及分式伯努利数的同余式。

On q-Euler numbers, q-Salie numbers and q-Carlitz numbers

Let $(a;q)_n=\prod_{0\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Salié numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows: $$\sum_{n=0}^{\infty}E_n(q)\frac{x^n}{(q,q)_n} =1/\sum_{n=0}^{\infty}\frac{q^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}S_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n}}{(q;q)_{2n}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n-1)}x^{2n}}{(q;q)_{2n}},$$ $$\sum_{n=0}^{\infty}C_n(q)\frac{x^n}{(q;q)_n} =\sum_{n=0}^{\infty}\frac{q^{n(n-1)}x^{2n+1}}{(q;q)_{2n+1}} /\sum_{n=0}^{\infty}\frac{(-1)^nq^{n(2n+1)}x^{2n+1}}{(q;q)…