scholarly journals Quadratic residue modulo a power of 2

2018 ◽  
Author(s):  
Thinh D. Nguyen
Keyword(s):  
Prime Power ◽  
Quadratic Residue ◽  
Residue Modulo ◽  
Quadratic Residues ◽  
Powers Of 2 ◽  
The Subject ◽  
Quadratic Residue Modulo

Quadratic residue modulo an odd prime power has been studied for centuries. Many mathematical tools have been devised to deal with those odd prime power. The left moduli of powers of 2 are thus naturally the subject of this article. We set the objectives of showing some intriguing properties of quadratic residues modulo an even prime power. Though humble in its significance, our results are achieved after years of reading Prasolov’s and I. F. Sharygin’s maths books.

scholarly journals Magic and Modulo Magicness of Paley Digraph

2019 ◽  
Vol 8 (2S11) ◽  
pp. 2850-2852
Keyword(s):  
Hadamard Matrices ◽  
Quadratic Residue ◽  
Round Robin ◽  
Residue Modulo ◽  
Quadratic Residues ◽  
Round Robin Tournaments ◽  
Modulo P ◽  
Quadratic Residue Modulo

A special digraph arises in round robin tournaments. More exactly, a tournament Tq with q players 1, 2, ... , q in which there are no draws. This gives rise to a digraph in which either (u, v) or (v, u) is an arc for each pair u, v. Graham and Spencer defined the tournament as, The nodes of digraph Dp are {0, 1, ... , p -1} and Dp contains the arc (u, v) if and only if u - v is a quadratic residue modulo p where p  3(mod 4) be a prime. This digraph is referred as the Paley tournament. Raymond Paley was a person raised Hadamard matrices by using this quadratic residues. So to honor him this tournament was named as Paley tournament. These results were extended by Bollobas for prime powers. Modular super edge trimagic labeling and modular super vertex magic total labeling has been investigated in this paper.

scholarly journals On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jingzhe Wang
Keyword(s):  
Exponential Sums ◽  
Quadratic Residue ◽  
Fourth Power ◽  
Power Mean ◽  
Residue Modulo ◽  
Fourth Power Mean ◽  
Elementary Methods ◽  
Quadratic Residue Modulo

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

scholarly journals Steiner’s Porism in finite Miquelian Möbius planes

2018 ◽  
Vol 18 (1) ◽  
pp. 55-68
Author(s):  
Norbert Hungerbühler ◽  
Katharina Kusejko
Keyword(s):  
Finite Fields ◽  
Prime Power ◽  
Euclidean Plane ◽  
Quadratic Residue ◽  
Residue Theorem ◽  
Common Tangent ◽  
Möbius Transformations ◽  
Concentric Circles ◽  
Mobius Transformations

AbstractWe investigate Steiner’s Porism in finite Miquelian Möbius planes constructed over the pair of finite fields GF(q) and GF(q2), for an odd prime powerq. Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner’s Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in GF(q). These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under Möbius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.

scholarly journals SOMEn−2TERRACES FROMnPOWER-SEQUENCES,nBEING AN ODD PRIME POWER

2009 ◽  
Vol 52 (1) ◽  
pp. 65-85
Author(s):  
IAN ANDERSON ◽  
D. A. PREECE
Keyword(s):  
Prime Power ◽  
Type A ◽  
Zero Element ◽  
Powers Of 2 ◽  
Power Sequence

AbstractA terrace formis an arrangement (a1,a2, . . . ,am) of themelements ofmsuch that the sets of differencesai+1−aiandai−ai+1(i= 1, 2, . . . ,m− 1) between them contain each element ofm\ {0} exactly twice. Formodd, many procedures are available for constructing power-sequence terraces form; each such terrace may be partitioned into segments, one of which contains merely the zero element ofm, whereas each other segment is either (a) a sequence of successive powers of an element ofmor (b) such a sequence multiplied throughout by a constant. We now adapt this idea by using power-sequences inn, wherenis an odd prime power, to obtain terraces form, wherem=n− 2. We write each element fromnso that they lie in the interval [0,n− 1] and then delete 0 andn− 1 so that they leaven− 2 elements that may be interpreted as the elements ofn−2. A segment of one of the new terraces may be of type (a) or (b), incorporating successive powers of 2, with each entry evaluated modulon. Our constructions providen−2terraces for all odd primesnsatisfying 0 <n< 1,000 except forn= 127, 241, 257, 337, 431, 601, 631, 673, 683, 911, 937 and 953. We also providen−2terraces forn= 3r(r> 1) and for some valuesn=p2, wherepis prime.

scholarly journals Symmetries of abelian Chern-Simons theories and arithmetic

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Time Reversal ◽  
Quadratic Residue ◽  
Complete Characterization ◽  
Simons Theory ◽  
Prime Factorization ◽  
Residue Modulo ◽  
The Matrix ◽  
Quantum Symmetries ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including U(1)k Chern-Simons theory and (ℤk)ℓ gauge theories. For example, we prove that U(1)k Chern-Simons theory is time-reversal invariant if and only if −1 is a quadratic residue modulo k, which happens if and only if all the prime factors of k are Pythagorean (i.e., of the form 4n + 1), or Pythagorean with a single additional factor of 2. Many distinct non-abelian finite symmetry groups are found.

On Residue Difference Sets

1953 ◽  
Vol 5 ◽  
pp. 425-432 ◽  
Author(s):  
Emma Lehmer
Keyword(s):  
Necessary Conditions ◽  
Difference Sets ◽  
Difference Set ◽  
Finite Geometries ◽  
Powers Of 2 ◽  
Higher Power ◽  
Modulo P ◽  
The Subject

In recent years the subject of difference sets has attracted a considerable amount of attention in connection with problems in finite geometries [4]. Difference sets arising from higher power residues were first discussed by Chowla [1], who proved that biquadratic residues modulo p form a difference set if (p — l )/4 is an odd square. In this paper we shall prove a similar result for octic residues and develop some necessary conditions which will eliminate all odd power residue difference sets and many others. We also prove that a perfect residue difference set (that is, one in which every difference appears exactly once) contains all the powers of 2 modulo p.

scholarly journals Doubly regular tournaments of Szekeres type

1982 ◽  
Vol 32 (3) ◽  
pp. 399-404 ◽  
Author(s):  
Noboru Ito
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Quadratic Residue ◽  
Residue Type ◽  
Regular Tournament

AbstractThe purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

scholarly journals Class numbers and quadratic residues

1976 ◽  
Vol 17 (1) ◽  
pp. 47-52 ◽  
Author(s):  
S. Chowla ◽  
J. Friedlander
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Strong Connection ◽  
Quadratic Residues ◽  
The Subject

It has long been known that there is a strong connection between the class numbers of quadratic fields and the distribution of quadratic residues. This connection is exemplified, for instance, by the class number formulae of Dirichlet, which have formed the basis of much of the work on the subject of class numbers.

scholarly journals A Survey of the Hadamard Maximal Determinant Problem

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Patrick Browne ◽  
Ronan Egan ◽  
Fintan Hegarty ◽  
Padraig Ó Catháin
Keyword(s):  
Historical Development ◽  
Design Theory ◽  
Infinite Family ◽  
Upper Bounds ◽  
Quadratic Residues ◽  
Maximum Value ◽  
Gram Matrices ◽  
The Subject ◽  
First Time ◽  
Determinant Theory

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructive lower bounds (achieved largely via quadratic residues in finite fields and concepts from design theory). To provide an impression of the historical development of the subject, we have attempted to modernise many of the original proofs, while maintaining the underlying ideas. Thus some of the proofs have the flavour of determinant theory, and some appear in print in English for the first time. We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving asymptotic analysis which has not previously appeared in the literature. We prove that there exists an infinite family of matrices achieving at least 0.48 of the maximal determinant bound. Previously the best known constant for a result of this type was 0.34.

Words

PMLA ◽  
1935 ◽  
Vol 50 (4) ◽  
pp. 1320-1327
Author(s):  
Colbert Searles
Keyword(s):  
Subject Matter ◽  
Assistant Professor ◽  
The Subject

THE germ of that which follows came into being many years ago in the days of my youth as a university instructor and assistant professor. It was generated by the then quite outspoken attitude of colleagues in the “exact sciences”; the sciences of which the subject-matter can be exactly weighed and measured and the force of its movements mathematically demonstrated. They assured us that the study of languages and literature had little or nothing scientific about it because: “It had no domain of concrete fact in which to work.” Ergo, the scientific spirit was theirs by a stroke of “efficacious grace” as it were. Ours was at best only a kind of “sufficient grace,” pleasant and even necessary to have, but which could, by no means ensure a reception among the elected.

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