Related papers: The tangent function and power residues modulo pri…

We completely calculate the $RO(\mathbb{Z}/p)$-graded coefficients $H\underline{\mathbb{Z}/p}_\star H\underline{\mathbb{Z}/p}$ for the constant Mackey functor $\underline{\mathbb{Z}/p}$.

In this paper, we first find the distribution of nth power residues modulo a prime $p$ by analyzing sums involving Dirichlet characters. We then extend this method to characterize the distribution of powers in arbitrary finite fields.

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $\nu_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $\nu_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at…

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that $p(n)$ is odd exactly $50\%$ of the time. Here, we greatly extend on our previous paper by providing a…

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…

We fix a prime number $p$ and $\K$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\Ko$ unramified outside $p$. The aim of this paper is to study the $\Z_p$-module $\gal(M/\Ko)$ and to give a method to effectively…

Let ${\Bbb Z}_{m}$ be the additive group of residue classes modulo $m$ and $s(m_{1},m_{2})$ denote the number of subgroups of the group ${\Bbb Z}_{m_{1}}\times {\Bbb Z}_{m_{2}}$, where $m_{1}$ and $m_{2}$ are arbitrary positive integers. We…

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd}…

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…

The finite ring Z_k = Z(+,.) mod p^k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units G_k in Z_k(.) has order (p-1)p^{k-1}, implying product structure G_k = A_k B_k. Here core A_k of order p-1 is an…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$;…

A study of products of factorials modulo p and complete residues.

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios…

In this paper we deduce some new supercongruences modulo powers of a prime $p>3$. Let $d\in\{0,1,\ldots,(p-1)/2\}$. We show that $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{2k}{k+d}}{8^k}\equiv 0\ (\mbox{mod}\ p)\ \ \ \mbox{if}\ d\equiv…

Let $\xi$ and $m$ be integers satisfying $\xi\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$…

Let $p$ be an odd prime, and let $m$ be an integer with $p\nmid m$. In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hbox{implies}\quad\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak…