Extension and Generalization of Fermat's Little Theorem to The
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Number Theory, Dover Publications, Inc., New York, 1994
Theory of Numbers Divisibility Theory in the Integers, The Theory of Congruences, Number-Theoretic Functions, Primitive Roots, Quadratic Residues Yotsanan Meemark Informal style based on the course 2301331 Theory of Numbers, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Second version August 2016 Any comment or suggestion, please write to [email protected] Contents 1 Divisibility Theory in the Integers 1 1.1 The Division Algorithm and GCD . 1 1.2 The Fundamental Theorem of Arithmetic . 5 1.3 The Euclidean Algorithm and Linear Diophantine Equations . 8 2 The Theory of Congruences 13 2.1 Basic Properties of Congruence . 13 2.2 Linear Congruences . 16 2.3 Reduced Residue Systems . 18 2.4 Polynomial Congruences . 21 3 Number-Theoretic Functions 25 3.1 Multiplicative Functions . 25 3.2 The Mobius¨ Inversion Formula . 28 3.3 The Greatest Integer Function . 30 4 Primitive Roots 33 4.1 The Order of an Integer Modulo n ............................ 33 4.2 Integers Having Primitive Roots . 35 4.3 nth power residues . 39 4.4 Hensel’s Lemma . 41 5 Quadratic Residues 45 5.1 The Legendre Symbol . 45 5.2 Quadratic Reciprocity . 48 Bibliography 53 Index 54 Chapter 1 Divisibility Theory in the Integers Let N denote the set of positive integers and let Z be the set of integers. 1.1 The Division Algorithm and GCD Theorem 1.1.1. [Well-Ordering Principle] Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a ≤ b for all b 2 S. -
Gaussian Prime Labeling of Super Subdivision of Star Graphs
of Math al em rn a u ti o c J s l A a Int. J. Math. And Appl., 8(4)(2020), 35{39 n n d o i i t t a s n A ISSN: 2347-1557 r e p t p n l I i c • Available Online: http://ijmaa.in/ a t 7 i o 5 n 5 • s 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications Gaussian Prime Labeling of Super Subdivision of Star Graphs T. J. Rajesh Kumar1,∗ and Antony Sanoj Jerome2 1 Department of Mathematics, T.K.M College of Engineering, Kollam, Kerala, India. 2 Research Scholar, University College, Thiruvananthapuram, Kerala, India. Abstract: Gaussian integers are the complex numbers of the form a + bi where a; b 2 Z and i2 = −1 and it is denoted by Z[i]. A Gaussian prime labeling on G is a bijection from the vertices of G to [ n], the set of the first n Gaussian integers in the spiral ordering such that if uv 2 E(G), then (u) and (v) are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs. MSC: 05C78. Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs. © JS Publication. 1. Introduction The graphs considered in this paper are finite and simple. The terms which are not defined here can be referred from Gallian [1] and West [2]. A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending upon the vertex labels f(x) and f(y). -
The Complete Splittings of Finite Abelian Groups
The complete splittings of finite abelian groups Kevin Zhaoa aDepartment of Mathematics, South China normal university, Guangzhou 510631, China Abstract Let G be a finite group. We will say that M and S form a complete split- ting (splitting) of G if every element (nonzero element) g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, and 0 has a such representation (while 0 has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our de- scription is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups. Keywords: complete splittings, splittings, cyclic groups, finite abelian groups, relationship. 1. Introduction The splittings of finite abelian groups are closely related to lattice tilings and are easy to be generated to lattice packings. Let K0 be a polytope composed of unit cubes and v + K0 be a translate of K0 for some vector v. A family of translations {v + K0 : v ∈ H} is called an integer lattice packing if H is an n-dimensional subgroup of Zn and, for any two vectors v and w in H, the interiors of v + K0 and w + K0 are disjoint; furthermore, if the n-dimensional Euclidean space Zn is contained in the family of these translations, then we call it an integer lattice tiling. The splitting problem can be traceable to a geometric problem posed by H. -
Number Theory Course Notes for MA 341, Spring 2018
Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . -
THE GAUSSIAN INTEGERS Since the Work of Gauss, Number Theorists
THE GAUSSIAN INTEGERS KEITH CONRAD Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = fa + bi : a; b 2 Zg: Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each section: (1) the norm on Z[i] (2) divisibility in Z[i] (3) the division theorem in Z[i] (4) the Euclidean algorithm Z[i] (5) Bezout's theorem in Z[i] (6) unique factorization in Z[i] (7) modular arithmetic in Z[i] (8) applications of Z[i] to the arithmetic of Z (9) primes in Z[i] 1. The Norm In Z, size is measured by the absolute value. In Z[i], we use the norm. Definition 1.1. For α = a + bi 2 Z[i], its norm is the product N(α) = αα = (a + bi)(a − bi) = a2 + b2: For example, N(2 + 7i) = 22 + 72 = 53. For m 2 Z, N(m) = m2. In particular, N(1) = 1. Thinking about a + bi as a complex number, its norm is the square of its usual absolute value: p ja + bij = a2 + b2; N(a + bi) = a2 + b2 = ja + bij2: The reason we prefer to deal with norms on Z[i] instead of absolute values on Z[i] is that norms are integers (rather than square roots), and the divisibility properties of norms in Z will provide important information about divisibility properties in Z[i]. -
Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases
INTERSECTIONS OF DELETED DIGITS CANTOR SETS WITH GAUSSIAN INTEGER BASES Athesissubmittedinpartialfulfillment of the requirements for the degree of Master of Science By VINCENT T. SHAW B.S., Wright State University, 2017 2020 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES May 1, 2020 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Vincent T. Shaw ENTITLED Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases BE ACCEPTED IN PARTIAL FULFILLMENT OF THE RE- QUIREMENTS FOR THE DEGREE OF Master of Science. ____________________ Steen Pedersen, Ph.D. Thesis Director ____________________ Ayse Sahin, Ph.D. Department Chair Committee on Final Examination ____________________ Steen Pedersen, Ph.D. ____________________ Qingbo Huang, Ph.D. ____________________ Anthony Evans, Ph.D. ____________________ Barry Milligan, Ph.D. Interim Dean, School of Graduate Studies Abstract Shaw, Vincent T. M.S., Department of Mathematics and Statistics, Wright State University, 2020. Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases. In this paper, the intersections of deleted digits Cantor sets and their fractal dimensions were analyzed. Previously, it had been shown that for any dimension between 0 and the dimension of the given deleted digits Cantor set of the real number line, a translate of the set could be constructed such that the intersection of the set with the translate would have this dimension. Here, we consider deleted digits Cantor sets of the complex plane with Gaussian integer bases and show that the result still holds. iii Contents 1 Introduction 1 1.1 WhystudyintersectionsofCantorsets? . 1 1.2 Prior work on this problem . 1 2 Negative Base Representations 5 2.1 IntegerRepresentations ............................. -
Primes in Arithmetical Progression
Colby College Digital Commons @ Colby Honors Theses Student Research 2019 Primes in Arithmetical Progression Edward C. Wessel Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Analysis Commons, and the Number Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Wessel, Edward C., "Primes in Arithmetical Progression" (2019). Honors Theses. Paper 935. https://digitalcommons.colby.edu/honorstheses/935 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Primes in Arithmetical Progression Edward (Teddy) Wessel April 2019 Contents 1 Message to the Reader 3 2 Introduction 3 2.1 Primes . 3 2.2 Euclid . 3 2.3 Euler and Dirichlet . 4 2.4 Shapiro . 4 2.5 Formal Statement . 5 3 Arithmetical Functions 6 3.1 Euler’s Totient Function . 6 3.2 The Mobius Function . 6 3.3 A Relationship Between j and m ................ 7 3.4 The Mangoldt Function . 8 4 Dirichlet Convolution 9 4.1 Definition . 10 4.2 Some Basic Properties of Dirichlet Multiplication . 10 4.3 Identity and Inverses within Dirichlet Multiplication . 11 4.4 Multiplicative Functions and their Relationship with Dirich- let Convolution . 13 4.5 Generalized Convolutions . 15 4.6 Partial Sums of Dirichlet Convolution . -
Gaussian Integers
Gaussian integers 1 Units in Z[i] An element x = a + bi 2 Z[i]; a; b 2 Z is a unit if there exists y = c + di 2 Z[i] such that xy = 1: This implies 1 = jxj2jyj2 = (a2 + b2)(c2 + d2) But a2; b2; c2; d2 are non-negative integers, so we must have 1 = a2 + b2 = c2 + d2: This can happen only if a2 = 1 and b2 = 0 or a2 = 0 and b2 = 1. In the first case we obtain a = ±1; b = 0; thus x = ±1: In the second case, we have a = 0; b = ±1; this yields x = ±i: Since all these four elements are indeed invertible we have proved that U(Z[i]) = {±1; ±ig: 2 Primes in Z[i] An element x 2 Z[i] is prime if it generates a prime ideal, or equivalently, if whenever we can write it as a product x = yz of elements y; z 2 Z[i]; one of them has to be a unit, i.e. y 2 U(Z[i]) or z 2 U(Z[i]): 2.1 Rational primes p in Z[i] If we want to identify which elements of Z[i] are prime, it is natural to start looking at primes p 2 Z and ask if they remain prime when we view them as elements of Z[i]: If p = xy with x = a + bi; y = c + di 2 Z[i] then p2 = jxj2jyj2 = (a2 + b2)(c2 + d2): Like before, a2 + b2 and c2 + d2 are non-negative integers. Since p is prime, the integers that divide p2 are 1; p; p2: Thus there are three possibilities for jxj2 and jyj2 : 1. -
Fermat Test with Gaussian Base and Gaussian Pseudoprimes
Czechoslovak Mathematical Journal, 65 (140) (2015), 969–982 FERMAT TEST WITH GAUSSIAN BASE AND GAUSSIAN PSEUDOPRIMES José María Grau, Gijón, Antonio M. Oller-Marcén, Zaragoza, Manuel Rodríguez, Lugo, Daniel Sadornil, Santander (Received September 22, 2014) Abstract. The structure of the group (Z/nZ)⋆ and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Eu- ler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider 2 2 the underlying group Gn := {a + bi ∈ Z[i]/nZ[i]: a + b ≡ 1 (mod n)}. In particular, we characterize Gaussian Carmichael numbers via a Korselt’s criterion and present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers n ≡ 3 (mod 4). There are also 18 no known composite numbers less than 10 in this family that are both pseudoprime to base 1 + 2i and 2-pseudoprime. Keywords: Gaussian integer; Fermat test; pseudoprime MSC 2010 : 11A25, 11A51, 11D45 1. Introduction Most of the classical primality tests are based on Fermat’s little theorem: let p be a prime number and let a be an integer such that p ∤ a, then ap−1 ≡ 1 (mod p). This theorem offers a possible way to detect non-primes: if for a certain a coprime to n, an−1 6≡ 1 (mod n), then n is not prime. -
Introduction to Abstract Algebra “Rings First”
Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals......................................... -
Number-Theoretic Transforms Using Cyclotomic Polynomials
MATHEMATICS OF COMPUTATION VOLUME 52, NUMBER 185 JANUARY 1989, PAGES 189-200 Parameter Determination for Complex Number-Theoretic Transforms Using Cyclotomic Polynomials By R. Creutzburg and M. Tasche Abstract. Some new results for finding all convenient moduli m for a complex number- theoretic transform with given transform length n and given primitive rath root of unity modulo m are presented. The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers. 1. Introduction. With the rapid advances in large-scale integration, a growing number of digital signal processing applications becomes attractive. The number- theoretic transform (NTT) was introduced as a generalization of the discrete Fourier transform (DFT) over residue class rings of integers in order to perform fast cyclic convolutions without roundoff errors [7, pp. 158-167], [10, pp. 211-216], [3]. The main drawback of the NTT is the rigid relationship between obtainable transform length and possible computer word length. In a recent paper [4], the authors have discussed this important problem of parameter determination for NTT's in the ring of integers by studying cyclotomic polynomials. The advantage of the later introduced (see [7, pp. 210-216], [9], [10, pp. 236-239], [5]) complex number-theoretic transforms (CNT) over the corresponding rational transforms is that the transform length is larger for the same modulus. In this note, we consider the problem of parameter determination for CNT, and we extend the results of [4] to the ring of Gaussian integers. 2. Primitive Roots of Unity Modulo m. By Z and I[i] we denote the ring of integers and the ring of Gaussian integers, respectively. -
GAUSSIAN INTEGERS 1. Basic Definitions a Gaussian Integer Is a Complex Number Z = X + Yi for Which X and Y, Called Respectively
GAUSSIAN INTEGERS 1. Basic Definitions A Gaussian integer is a complex number z = x + yi for which x and y, called respectively the real and imaginary parts of z, are integers. In particular, since either or both of x and y are allowed to be 0, every ordinary integer is also a Gaussian integer. We recall that the complex conjugate of z is defined by z = x iy and satisfies, for arbitrary complex numbers z and w, z + w = z−+ w and zw = zw. We remark that sums, products, and complex conjugates of Gaussian Integers are again Gaussian integers. For any complex number z, we define the norm of z by N(z) = zz = x2 +y2, where x and y are the respectively the real and imaginary parts of z. It follows that N(zw) = N(z)N(w). We remark that the norm of any complex number is a non-negative real number, the norm of a Gaussian integer is a non-negative integer, and only 0 has norm 0. We observe further that only 1 and i have norm 1. These are called unit Gaussian integers, or± units, and± two Gaussian integers are called associates if they can be obtained from one another by multiplication by units. Note that, in general, z and its associates are distinct from z and its associates. The exceptions to this rule occur when the real and imaginary parts of z have the same absolute value, and when the real or imaginary part of z is 0. We defined divisibility for the Gaussian integers exactly as for inte- gers.