Related papers: Supercongruences concerning truncated hypergeometr…

This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|…

Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain…

In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\sum_{r=0}^{n-1}\frac1{1+\sin2\pi\frac{x+r}n+\cos2\pi\frac{x+r}n}…

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

For $k\ge 3$ and $\epsilon>0$, let $H$ be a $k$-partite $k$-graph with parts $V_1,\dots, V_k$ each of size $n$, where $n$ is sufficiently large. Assume that for each $i\in [k]$, every $(k-1)$-set in $\prod_{j\in [k]\setminus \{i\}} V_i$…

Let ${{\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as…

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…

We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv…

Suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We denote $\gamma_1=\inf\limits_{u\in W^{1,…

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…

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n)…

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a…

Using Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3…

The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}\ (\gamma\neq 0,-1,-2,\cdots)$, which are of many properties and great…

The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{\Gamma(\frac{n}{2}-\frac{s}{p-1})\Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1})\Gamma(\frac{n-2s}{2}-\frac{s}{p-1})}…

We holographically study supersymmetric deformations of $N=3$ and $N=1$ superconformal field theories (SCFTs) in three dimensions using four-dimensional $N=4$ gauged supergravity coupled to three-vector multiplets with non-semisimple…

Fix an odd prime $p$. In this article, we provide a $\mathrm{mod}\ p$ harmonic number identity, which appears naturally in the canonical expansion of a root $\zeta_{p^n}$ of the $p^n$-th cyclotomic polynomial $\Phi_{p^n}(T)$ in the $p$-adic…

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$…

Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the…