scholarly journals Special Primes And Some Of Their Properties

2021 ◽  
Author(s):  
Anantha Krishna B ◽  
Mantha Sai Gopal ◽  
Sourangshu Ghosh
Keyword(s):  
Prime Number ◽  
Open Problem ◽  
Mod P

In this paper, we present the definition, some properties, and solve a problem on special primes. These properties help in providing us with a better understanding of the problem posed related to special primes on the open problem garden website. The problem involves finding all the primes q, given a prime p such that q≡1(mod p) and 2^((q−1)/p)≡1(mod q). We prove that a prime number q is a special prime of p if and only if the order of 2 in U(q) divides q−1p. Also, we prove that a prime number q is not a special prime for any prime number if 2 is a generator of the group U(q) and that there exist infinitely many special primes for any given prime number.

SIMILARITY INDICES OF AMPHICHEIRAL KNOTS AND TANGLES

2008 ◽  
Vol 17 (04) ◽  
pp. 483-494
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Open Problem ◽  
Necessary Conditions ◽  
Similarity Index ◽  
Vassiliev Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Similarity Indices ◽  
Mod P

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not. Let T and S be two tangles which are n-similar for some natural number n and let the closure [Formula: see text] of T be well-defined. Let T* and S* be the mirror images of T and S respectively. Then we show that for any prime number p, [Formula: see text] mod p for any integral Vassiliev invariant v of degree ≤ np. We also show that [Formula: see text] for any Vassiliev invariant w of degree ≤ n if n is odd. Therefore, if an amphicheiral knot can be distinguished from a trivial knot by a Vassiliev invariant, then it has an even triviality index. From these, we get some necessary conditions for a knot invariant to be a Vassiliev invariant and get a method to detect the similarity index of two knots or tangles.

An answer to a conjecture on the sum of element orders

2020 ◽  
pp. 2250067
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi ◽  
Morteza Jafarpour
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Nilpotent Group ◽  
Prime Number ◽  
Lower Bounds ◽  
Finite Groups ◽  
Open Problem ◽  
Element Orders ◽  
Equality Condition ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

scholarly journals Distribution of special sequences modulo a large prime

2003 ◽  
Vol 2003 (50) ◽  
pp. 3189-3194 ◽  
Author(s):  
M. Z. Garaev ◽  
Ka-Lam Kueh
Keyword(s):  
Prime Number ◽  
Primitive Root ◽  
Mod P

We study the sets{gx−gy(mod p):1≤x,y≤N}and{xy:1≤x,y≤N}wherepis a large prime number,gis a primitive root, andp2/3<N<p.

scholarly journals Differences between powers of a primitive root

2002 ◽  
Vol 29 (6) ◽  
pp. 325-331 ◽  
Author(s):  
Marian Vâjâitu ◽  
Alexandru Zaharescu
Keyword(s):  
Prime Number ◽  
Primitive Root ◽  
Mod P

We study the set of differences{gx−gy(modp):1≤x,   y≤N}wherepis a large prime number,gis a primitive root(modp), andp2/3<N<p.

CLASSIFICATION OF MINIMAL NONASSOCIATIVE MOUFANG LOOPS OF ORDER pq3

2013 ◽  
Vol 23 (08) ◽  
pp. 1895-1908 ◽  
Author(s):  
WING LOON CHEE ◽  
STEPHEN M. GAGOLA ◽  
ANDREW RAJAH
Keyword(s):  
Abelian Group ◽  
Open Problem ◽  
Related Question ◽  
Moufang Loop ◽  
The Third ◽  
Odd Order ◽  
Moufang Loops ◽  
Mod P

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q3can be constructed by using a non-abelian group of order q3. In [Moufang loops of odd order pq3, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq3exists if and only if q ≡ 1 ( mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq3for primes p < q.

Witt Rings and Permutation Polynomials

2005 ◽  
Vol 12 (01) ◽  
pp. 161-169 ◽  
Author(s):  
Qifan Zhang
Keyword(s):  
Prime Number ◽  
Permutation Polynomials ◽  
Polynomial Functions ◽  
Witt Rings ◽  
Orthogonal Systems ◽  
Mod P

Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over ℤ/p2ℤ and 3-tuples of polynomial functions over ℤ/pℤ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over ℤ/plℤ.

The least quadratic non-residue

2021 ◽  
pp. 205-231
Author(s):  
Kevin McGown ◽  
Enrique Treviño
Keyword(s):  
Prime Number ◽  
Content Type ◽  
Mathematical History ◽  
Survey Paper ◽  
Residue Modulo ◽  
History Of ◽  
Main Ideas ◽  
Mod P

For a prime number p p , we say a a is a quadratic non-residue modulo p p if there is no integer x x such that x 2 ≡ a mod p x^2\equiv a\bmod {p} . The problem of bounding the least quadratic non-residue modulo p p has a rich mathematical history. Moreover, there have been recent results, especially concerning explicit estimates. In this survey paper we give the history of the problem and explain many of the main achievements, giving explicit versions of these results in most cases. The paper is intended as a self-contained collection of the main ideas that have been used to attack the problem.

The mod p cohomology group of extra-special p-group of order p5 and of exponent p2

1996 ◽  
Vol 120 (3) ◽  
pp. 423-440 ◽  
Author(s):  
Pham Anh Minh
Keyword(s):  
Prime Number ◽  
Cyclic Group ◽  
Cohomology Group ◽  
Cohomology Groups ◽  
Central Product ◽  
Mod P

Let p be an odd prime number, and let M2 be the extra-special p-group of order p5 and of exponent p2. For every p-group K, we denote by H*(K) the mod p cohomology of K. The purpose of this paper is to calculate the mod p cohomology groups of M2 and of Cp2* M2 (the central product of the cyclic group Cp2 of order p2 and M2).

Images of mod p Galois Representations Associated to Elliptic Curves

2001 ◽  
Vol 44 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Amadeu Reverter ◽  
Núria Vila
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Torsion Points ◽  
Galois Action ◽  
Mod P

AbstractWe give an explicit recipe for the determination of the images associated to the Galois action on p-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over without complex multiplication with conductor less than 200 and for each prime number p.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

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