Zhi-Hong Sun
Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.
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For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that $P_{p-1}(a,x)\e (-1)^{<a>_p}P_{p-1}(a,-x)\mod {p^2}$, where $<a>_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$ and $\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$, where $[x]$ is the great…
Let $p\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$. We consider supersingular elliptic curves with a basis of $2$-torsion over $\overline{\mathbf{F}}_p$, or equivalently supersingular Legendre $λ$-invariants. Let $F_p(X,Y) \in \mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $λ$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\frac{F_p(X,X^{p})}{p}$ is in $\mathbf{Z}[X]$. We give a formula for $R(λ)$ if $λ$ is a supersingular. This formula is related to the Manin--Dri…
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that $\sum_{k=0}^{\frac{p-1}{2}}\binom{2k}k^3\equiv 0\pmod {p^2}$ for $p\equiv 3,5,6\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\frac {p}{4}]}(t)\equiv -\big(\frac{-6}{p}\big)\sum_{x=0}^{p-1} \big(\frac{x^3-3/2(3t+5)x-9t-7}{p}\big)\pmod p$ and d…
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\e 0\pmod p$, $$\align &P_{[\frac p6]}(t) \e -\Big(\frac 3p\Big)\sum_{x=0}^{p-1}\Big(\frac{x^3-3x+2t}p\Big)\pmod p, &\Big(\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big)\Big)^2\equiv \Big(\frac{-3m}p\Big) \sum_{k=0}^{[p/6]}\binom{2k}k\binom{3k}k\binom{6k}{3k} \Big(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\Big)^k\pmod p,$$ where $(\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer…
Let $p>3$ be a prime and $m,n\in\Bbb Z$ with $p\nmid mn$. Built on the work of Morton, in the paper we prove the uniform congruence: $$&\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big) \equiv {-(-3m)^{\frac{p-1}4} \sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\t{if $4\mid p-1$,} \frac{2m}{9n}(\frac{-3m}p)(-3m)^{\frac{p+1}4} \sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\text{if $4\mid p-3$,}$$ where $(\frac ap)$ is the Legendre symbol. We also establish many congruences for $x\pmod p$, where $x$ is given …
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 $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.
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。
Let $P,Q\in\Bbb Z$, $U_0=0,\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. In this paper we obtain a general congruence for $U_{kmn^r}/U_k\pmod {n^{r+1}}$, where $k,m,n,r$ are positive integers. As applications we extend Lucas' law of repetition and characterize the square prime factors of $a^n+1$ or $S_n$, where $\{S_n\}$ is given by $S_1=P^2+2$ and $S_{k+1}=S_k^2-2\ (k\ge 1)$.
Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac p{k+b}\pmod {p^2}$. For $n=0,1,2,\ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$\sum_{n=0}^{p-1}\frac{D_n}{16^n},\quad \sum_{n=0}^{p-1}\frac{D_n}{4^n}, \quad \sum_{n=0}^{p-1}\frac{b_n}{(-3)^n},\quad \sum_{n=0}^{p-1}\frac{b_n}{(-27)^n}\pmod {p^2}$$ in terms of certain binary quadratic forms.
We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.
Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.
Congruences concerning generalized central trinomial coefficients via the constant term method
研究广义中心三项式系数，利用常数项方法和生成函数证明参数同余式，证实了孙志伟的几个猜想。
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.
In this paper, we mainly prove the following congruence conjectured by J.-C. Liu: $$ {}_6F_5\bigg[\begin{matrix}\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\&\frac{1}{4}&1&1&1&1\end{matrix}\bigg|\ -1\bigg]_{\frac{p-1}{2}}\equiv-\frac{p^3}{16}Γ_p\left(\frac{1}{4}\right)^4\pmod{p^5}, $$ where $p\geq5$ are primes with $p\equiv3\pmod{4}$.
Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.
For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400].
Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.
We prove a Ramanujan-type formula for $520/π$ conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for $1/π$.
Let $n$ be a nonnegative integer. The $n$-th Apéry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime $p\geq7$ $$ \sum_{k=0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $$ and for any prime $p\geq5$ $$ \sum_{k=0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $$ where $H_n=\sum_{k=1}^n1/k$ denotes the $n$-th harmonic number and $B_0,B_1,\ldots$ are the well-known Bernoulli numbers. In this…
Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{(-144)^k}\ \text{and} \ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{648^k}$$ modulo $p^2$, and partial results for $\sum_{k=0}^{(p-1)/2} \binom{2k}k^3\frac{w(k)}{m^k}$ modulo $p^2$, where $m\in\{1,16,-64,256,-512,4096\}$ and $w(k)\in\{k^2,k^3,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3}, \frac 1{2k-1},\frac 1{k+2}\}$.
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.
In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\sum_{k=0}^{p^a-1}\binom{2k}{k+δ}$ mod $p^3$ for $δ=0,1$. We also show that $$C_n^{-1}\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol.
Newform Eisenstein Congruences of Local Origin
本文提出一个关于新形式与新艾森斯坦级数之间艾森斯坦同余的普遍猜想，并探讨其与布洛赫-卡托猜想的关系。
Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Tay…
In this paper, we partially extend recent results of Wan concerning the relationship between the zeta functions of a Calabi-Yau hypersurface and its (singular) mirror variety.
We prove two results concerning the generalized Fermat equation $x^4+y^4=z^p$. In particular we prove that the First Case is true if $p \neq 7$.
Sun-type determinant and permanent congruences
本文统一处理Sun提出的关于模素数剩余类上行列式和永久式的猜想，通过行列式论证、Cauchy核和Cayley核的永久式恒等式等工具，证明了多个猜想并给出了p^2和p^3模的同余式。
A Determinant Congruence Conjectured by Sun
本文证明了孙氏关于二元二次型行列式的一个加强形式猜想，通过分析范德蒙德因子和模p矩阵的秩，得到了复合数和素数情况下的同余结论。
Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary $k$-th power residue modulo $p$ and primitive roots of a power of $p$. Finally, we discuss some permutation problems con…
Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $α$ and $m\in\{1,2,3\}$, we have \begin{align*} \sum_{k=0}^{p^αn-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\equiv\bigg(\frac{m(m-4)}{p}\bigg)\sum_{k=0}^{p^{α-1}n-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\pmod{p^{2α}}, \end{align*} where $(x)_k=x(x+1)\cdots(x+k-1)$ and $\big(\frac{\cdot}{\cdot}\big)$ denotes the Legendre symbol. Also, when $m=4$, \begin{align*} \sum_{k=0}^{p^αn-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\equiv p\sum_{k=0}^{p^{α-1}n-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\pmod{p^{2α}}. \end{align*…
On $p$-adic congruences involving $\sqrt d$
针对奇素数$p$和不被$p$整除的整数$d$，证明了含$\sqrt{d}$的特定乘积多项式模$p$的同余式（分Legendre符号为1和-1两种情况），推广N. Kalinin猜想，并得到Wolstenholme型同余式。
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$.
The congruence subgroup property for $S$-arithmetic subgroups of simple algebraic groups when $S$ has positive Dirichlet density
本文研究了当S具有正狄利克雷密度时，简单代数群的S-算术子群的同余子群性质，证明了同余核为平凡群，为塞雷的同余子群猜想提供了证据。
Two determinant evaluations in Sun's conjectures involving Legendre symbols
通过约简四元行列式到相邻余子式的四元逆包，证明了Sun关于Legendre符号矩阵猜想中的第一个行列式求值，并给出了第二个行列式模p的一致精确同余式。
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{3k}km^{-k}\mod {p^2}$.
Congruences modulo powers of $3$ for $6$-colored generalized Frobenius partitions
本文研究6色广义弗罗贝尼乌斯划分函数cϕ_6(n)的同余性质，采用Banerjee和Smoot发展的方法验证了Gu等人提出的猜想。
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.
The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3 &\equiv p\left(\frac{-4x-3}{p}\right) \pmod{p^2}, \\ \sum_{k=0}^{p-1}(2k+1)D_k(x)^4 &\equiv p \pmod{p^2}, \\ \sum_{k=0}^{p-1}(-1)^k(2k+1)D_k(x)^3 &\equiv p\left(\frac{4x+1}{p}\right) \pmod{p^2}, \end{align*} where $\big(\frac{\cdot}{p}\big)$ denotes the Legendre symbol. The first two congruences confirm a conjecture of Z.-W. Sun [Sci. China 57 (2014), 1375--1400].…
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…
A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields
本文研究有限域乘法子群相关矩阵的行列式，证明了二次平方元矩阵的显式行列式公式，从而证实Zhi-Wei Sun的猜想。
In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `Generalized Perrin-Riou Conjecture', predicts a precise congruence relation between a `Darmon-type derivative' of the zeta element of $E$ over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the $L$-function of $E$ over $\mathbb{Q}$. We prove that the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou's conjecture on the logarithm of Kato's …
A family of polynomials and related congruences and series
本文研究了一族多项式及其相关同余与级数，提出了关于$1/π$的开放猜想。
Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $π^2$, $π^{-2}$ …
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.
A new kind of numbers and related congruences
研究整数\(l>0\)、\(m\geqslant0\)相关的数\(S_l^{(m)}(n)\)，通过证明得出特定素数模下的同余式，尤其当\(l = 4\)、\(m = 2\)时得到关于\(Domb\)数\(D(n)\)的奇特素数模同余式。
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…
Real exponential sums over primes and prime gaps
研究证明在短区间内存在素数，解决了长期存在的问题，并验证了关于素数分布的旧猜想。
We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.
Cohomology of Congruence Subgroups of SL_4(Z). III
本文通过稀疏矩阵约简的新方法，将SL_4(Z)的同余子群Γ_0(N)在素数水平N≤211上的上同调群H^5(Γ_0(N); C)的计算扩展到更大范围，并提出了关于Eisenstein级数和Siegel模形式贡献的两个猜想。
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…
Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big…
In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(…
Proof of Two Conjectural $q$-Supercongruences Related to Van Hamme's $\mathrm{(G.2)}$ Supercongruence
本文证明了与Van Hamme的(G.2)超同余相关的两个q-超同余猜想，即Tang的猜想和Wang与Xu的猜想，完成了同余式的扩展与验证。
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q…
On a polynomial involving roots of unity and its applications
本文推广了 Gauss、Dirichlet 和 Sun 关于涉及单位根的多项式 $S_p(x)$ 的研究，并将其应用于扩展 S. Chowla 关于实二次域基本单位数的同余结果，同时给出了扩展 Ankeny-Artin-Chowla 猜想的等价形式。
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