Related papers: Congruences with intervals and subgroups modulo a …

This article is an overview of Zagier's and Kim's work on traces of singular moduli. We give more detailed or new proofs to some of their results and also describe some algorithms to compute spaces of Jacobi forms and weight $3/2$ modular…

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact…

We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.

For $m,q \in \mathbb{N}$, we call an $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$ good if there are infinitely many consecutive primes $p_1,\ldots,p_m$ satisfying $p_i \equiv a_i \pmod{q}$ for all $i$. We…

In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…

Let $R$ be a commutative ring with identity. For a finitely generated $R$-module $M$, the notion of associated prime submodules of $M$ is defined. It is shown that this notion inherits most of essential properties of the usual notion of…

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\…

For a prime $p>3$ and $a\in \Bbb Z$ with $p\nmid a$ let $V_p(x^2+\frac ax)$ be the residue-counts of $x^2+\frac ax$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we obtain an explicit formula for $V_p(x^2+\frac ax)$, which is…

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led…

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta},…

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…

In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

In the recent article arXiv:1606.03351, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end they…

Paul Erdos posed the following question: Is there a prime number $p>5$ such that the residues of $2!$, $3!$,\ldots, $(p-1)!$ modulo $p$ all are distinct. In this short note, we give the negative answer on this question in an elementary way.

We completely characterise when the sequence of free subgroup numbers of a finitely generated virtually free group is ultimately periodic modulo a given prime power.

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones…

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are…

We establish an elementary, but rather striking pattern concerning the quartic residues of primes $p$ that are congruent to 5 modulo 8. Let $g$ be a generator of the multiplicative group of $\mathbb Z_p$ and let $M$ be the $4\times 4$…

We investigate the existence of primes $p > 5$ for which the residues of $2!$, $3!$, \dots, $(p-1)!$ modulo $p$ are all distinct. We describe the connection between this problem and Kurepa's left factorial function, and report that there…