A variation of $q$-Wolstenholme's theorem

Wilson's and Wolstenholme's Theorems

We provide a proof of Wilson's Theorem and Wolstenholme's Theorem based on a direct approach by Lagrange requiring only basic properties of the primes and the Binomial theorem. The goal is to show how similar the two theorems are by providing the easiest proof possible in a single unified argument.

Wolstenholme 定理在 Gaussian 整数上的推广

Wolstenholme's theorem over Gaussian integers

本文将 Wolstenholme 定理推广至 Gaussian 整数环，证明互素 Gaussian 整数倒数和模 $p^4$ 为零，扩展至高次幂和，并以小素数计算验证及提出二项式系数相关猜想。

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 …

超越Wolstenholme定理

Beyond Wolstenholme's Theorem

本文研究Wolstenholme型求和，证明双积、三积及比值同余于二项式系数表达式，并推广至任意n项，联系多项式系数。

Variations of Lucas' Theorem Modulo Prime Powers

Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem modulo prime powers holds $$ {np^k +n_0 \choose mp^k+m_0}\equiv{np^{\lfloor(k-1)/3\rfloor} \choose mp^{\lfloor(k-1)/3\rfloor}} {n_0 \choose m_0} \pmod{p^k}. $$ In the proof the authors used their earlier result that present a generalized version of Lucas' Theorem. In this paper we present a a simple inductive proof of the above congruence. Our proof is based o…

Wilson's Theorem for Finite Fields

In this short note, we introduce an analogue of Wilson's theorem for all nonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with $|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leq q-1}a_{i_1}a_{i_2}... a_{i_k}=\left\lfloor\frac{k}{q-1}\right\rfloor(-1)^q\hspace{10mm}(k=1,2,..., q-1), $$ which the left hand side of above formula is the $k-$th elementary symmetric polynomial evaluated at $a_1,a_2,...,a_{q-1}$. Specially, letting $\mathbb{F}=\mathbb{Z}_p$ with $p\geq 3$, reproves Wilson's theorem and yields some Wilson type identities. Finally, we obtain a…

Variations on theorems of Mertens

We present variations on theorems of Mertens as special cases of Density Hypothesis. Moreover, we study a Serre's estimate concerning Lang-Weil estimate.

The $p$-adic variation of the Gross-Kohnen-Zagier theorem

We relate $p$-adic families of Jacobi forms to Big Heegner points constructed by B. Howard, in the spirit of the Gross-Kohnen-Zagier theorem. We view this as a GL(2) instance of a $p$-adic Kudla program.

An alternative proof of Sylvester's theorem and variations for more primes

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger versions of Sylvester's theorem and Bertrand's postulate with more primes, as well as an approach to get more results in this direction.

Variations of the Primitive Normal Basis Theorem

The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $α\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of $\mathbb F_{q^n}$ with multiplicative order $(q^n-1)/2$ and prescribed trace over $\mathbb F_q$.

A Polynomial Variation on Meinardus' Theorem

We develop a polynomial analogue of Meinardus' Thoerem for bivariate Euler products and apply it to the study of complex multiplicatively weighted partitions.

Variations of Landau's theorem for p-regular and p-singular conjugacy classes

The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as well as $p$-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of $p$-regular classes or the number of $p$-singular classes of the group. In particular, if $G$ is a finite group with $O_p(G)=1$ then $|G/F(G)|_{p'}$ is bounded in terms of the number of $p$-regular classes of $G$. However, …

Variations on a theorem of Tate

Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $Gal(\bar{F}/F) \to PGL_n(C)$ lift to $GL_n(C)$. We take special interest in the interaction of this result with algebraicity (on the automorphic side) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison …

Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain

Let $L^2(X,Σ,μ,τ)$ be a measure-preserving system, with $τ$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} τ^{P(n)} f \] converge pointwise for $f \in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.