scholarly journals On the number of unit solutions of cubic congruence modulo $ n $

2021 ◽  
Vol 6 (12) ◽  
pp. 13515-13524
Author(s):  
Junyong Zhao ◽  
◽  
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Linear Congruence ◽  
Residue Classes ◽  
Congruence Modulo ◽  
Quadratic Case

<abstract><p>For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document} </tex-math></disp-formula></p> </abstract>

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

On the addition of two weighted squares of units mod n

2016 ◽  
Vol 12 (07) ◽  
pp. 1783-1790 ◽  
Author(s):  
Cui-Fang Sun ◽  
Zhi Cheng
Keyword(s):  
Positive Integer ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Quadratic Congruence

For any positive integer [Formula: see text], let [Formula: see text] be the ring of residue classes modulo [Formula: see text] and [Formula: see text] be the group of its units. Recently, for any [Formula: see text], Yang and Tang obtained a formula for the number of solutions of the quadratic congruence [Formula: see text] with [Formula: see text] units, nonunits and mixed pairs, respectively. In this paper, for any [Formula: see text], we give a formula for the number of representations of [Formula: see text] as the sum of two weighted squares of units modulo [Formula: see text]. We resolve a problem recently posed by Yang and Tang.

On the Structure of G(H, k)

2014 ◽  
Vol 21 (02) ◽  
pp. 317-330 ◽  
Author(s):  
Guixin Deng ◽  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Explicit Formula ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Necessary And Sufficient

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

scholarly journals RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

2021 ◽  
Vol 9 (VII) ◽  
pp. 2598-2601
Author(s):  
Prof. B. M. Roy
Keyword(s):  
Positive Integer ◽  
Integer Multiple ◽  
Composite Modulus ◽  
First Case ◽  
Congruence Modulo ◽  
Quadratic Congruence ◽  
First Time

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

scholarly journals On a Double Series of Chan and Ong

2006 ◽  
Vol 13 (4) ◽  
pp. 793-805
Author(s):  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Double Series

Abstract An arithmetic identity is used to prove a relation satisfied by the double series . As an application an explicit formula is given for the number of representations of the positive integer 𝑛 by the form .

On the distribution of zeros of derivatives of the Riemann ξ-function

2020 ◽  
Vol 32 (1) ◽  
pp. 1-22
Author(s):  
Amita Malik ◽  
Arindam Roy
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Real Number ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Density Estimate ◽  
Distribution Of Zeros ◽  
Riemann Zeta ◽  
Derivatives Of

AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

ON CONJUGATES OF COLLATZ-TYPE MAPPINGS

2008 ◽  
Vol 04 (01) ◽  
pp. 117-120
Author(s):  
STEFAN KOHL
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Affine Mapping ◽  
Absolute Value ◽  
Residue Classes ◽  
Almost Everywhere ◽  
Finite Set ◽  
The Absolute ◽  
Almost All ◽  
Original Motivation

A mapping f : ℤ → ℤ is called residue-class-wise affine if there is a positive integer m such that it is affine on residue classes (mod m). If there is a finite set S ⊂ ℤ which intersects nontrivially with any trajectory of f, then f is called almost contracting. Assume that f is a surjective but not injective residue-class-wise affine mapping, and that the preimage of any integer under f is finite. Then f is almost contracting if and only if there is a permutation σ of ℤ such that fσ = σ-1 ◦ f ◦ σ is either monotonically increasing or monotonically decreasing almost everywhere. In this paper it is shown that if there is no positive integer k such that applying f(k) decreases the absolute value of almost all integers, then σ cannot be residue-class-wise affine itself. The original motivation for the investigations in this paper comes from the famous 3n + 1 Conjecture.

ON m-COVERS AND m-SYSTEMS

2009 ◽  
Vol 81 (2) ◽  
pp. 223-235
Author(s):  
ZHI-WEI SUN
Keyword(s):  
Positive Integer ◽  
Cyclotomic Fields ◽  
Residue Classes ◽  
Fixed Positive Integer

AbstractLet 𝒜={as(mod ns)}ks=0 be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results related to the covering multiplicity of 𝒜. In particular, we show that if every integer lies in more than m0=⌊∑ ks=11/ns⌋ members of 𝒜, then for any a=0,1,2,… there are at least ${m_0\choose \lfloor a/n_0\rfloor }$ subsets I of {1,…,k} with ∑ s∈I1/ns=a/n0. We also characterize when any integer lies in at most m members of 𝒜, where m is a fixed positive integer.

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