

 Jeffrey Gold
 2 years ago
 Views:
Transcription
1 Chapter 2 Remodulization of Congruences Proceedings NCUR VI. è1992è, Vol. II, pp. 1036í1041. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction Remodulization introduces a new method applied to congruences and systems of congruences. We prove the Chinese Remainder Theorem using the remodulization method and establish an eæcient method to solve linear congruences. The following is an excerpt of Remodulization of Congruences and its Applications ë2ë. Deænition 1 If a and b are integers, then a mod b = fa; a æ b; a æ 2b;:::g. We write x ç amodb, meaning that x is an element of the set a mod b. The common terminology is to say that x is congruent toa modulo b. These sets are frequently called residue classes since they consist of those numbers which, upon division by b, leave a remainder èresidueè of a. 1
2 CHAPTER 2. REMODULIZATION OF CONGRUENCES 2 Deænition 2 If a1;a2;::: ;a n ;b are integers, then ëa1;a2;::: ;a n ëmodb =èa1 mod bè ë èa2 mod bè ëæææëèa n mod bè : Theorem 1 Suppose a, b, andc are integers and cé0, then a mod b =ëa; a + b;::: ;a+èc, 1èbë modcb : Proof. We write a mod b = f a, 2cb; a, è2c, 1èb; ::: a, èc +1èb a, cb; a, èc, 1èb; ::: a, b a; a + b; ::: a +èc, 1èb; a + cb; a +èc +1èb; ::: a +è2c, 1èb; a +2cb; a +è2c +1èb; ::: a +è3c, 1èb; g and rewriting the rows a mod b = f a, 2cb; a + b, 2cb; ::: a +èc, 1èb, 2cb a, cb; a + b, cb; ::: a +èc, 1èb, cb a; a + b; ::: a +èc, 1èb; a + cb; a + b + cb; ::: a +èc, 1èb + cb; a +2cb; a + b +2cb; ::: a +èc, 1èb +2cb; g Then, forming unions on the extended columns, the result follows. We refer to this process as remodulization by a factor c. Suppose it is desired to express 1 mod 2 in terms of modulo 8, then, 1 mod 2 is remodulized by the factor 4, i.e., 1mod2=ë1; 3; 5; 7ë mod 8 : It is convenient to create a notation for the expression We write it as Reindexing the symbol S, ë a; a + b;::: ;a+èc, 1èbë modcb : c,1 ë ëa + kbë modcb : k=0 cë a mod b = ëèa, bè+kbë modcb : The Chinese Remainder Theorem ærst appeared in the ærst century A.D. The Chinese mathematician SunTsçu sought a solution to the following problem:
3 CHAPTER 2. REMODULIZATION OF CONGRUENCES 3 What numbers n, when divided by 3, 5, and 7, have remainders 2, 3, and 2, respectively? This problem also appeared in the Introductio Arithmeticae, written by Nicomachus of Gerasa, a Greek mathematician circa 100 A.D. The problem asks one to ænd the solution to a system of congruences èyè 8 é é: x ç a1 mod b1 x ç a2 mod b2. x ç a n mod b n where 0 ç a j éb j, and the b j are pairwise relatively prime. The idea is to remodulize each congruence in order to obtain a common modulus, thereby making the solution set the intersection of the resulting classes. These can be determined by direct observation of the sets of residues in the remodulized forms. Since the b j are pairwise relatively prime, the smallest common modulus is the product of the b j ; therefore we remodulize the j th congruence by the factor Performing these operations gives: 1 b k = c j ; then b j c j = C = b j x j ç 1 b j ë b k Simplifying the notation, we ænd ëèa j, b j è+b j mëmod c ë j x j ç ëèa j, b j è+b j mëmodc : The solution set to èyè is the intersection of the sets of initial elements mod C, i.e., èzè në x ç j=1 " cjë b k b k ëèa j, b j è+b j mëmodc Thus, for the original problem of SunTsçu, we have: 8 é : x ç 2mod3 x ç 3mod5 x ç 2mod7 è :
4 CHAPTER 2. REMODULIZATION OF CONGRUENCES 4 Since the b j are prime, the least common modulus is 3 æ 5 æ 7=105. The congruences are then remodulized by 35, 21, and 15, respectively. The resulting remodulizations are ë2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104ë mod 105 ; ë3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98, 103ë mod 105 ; and ë2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93, 100ë mod 105, where the intersection, 23 mod 105, is the complete solution set among the integers. This ultimately raises the question as to whether èzè në x ç j=1 " cjë ëèa j, b j è+b j mëmodc always contains exactly one element, given that 0 ç a j é b j and the b j are pairwise relatively prime. In the same example, if we remodulize to the product 105, we ænd that the solution set corresponding to the ærst two congruences ç x ç 2mod3 x ç 3mod5 is characterized as ë8; 23; 38; 53; 68; 83; 98ë mod 105, which does not appear to be ëunique"; however, this is equivalent to 8 mod 15, which, in the example given, has been remodulized by the factor 7. The solution 8 mod 15 is obtained by solving the ærst two congruences directly by the method described. As it happens, if one uses the smallest possible modulus, the answer to our question is yes. è Theorem 2 èchineseremaindert heoremè The system èyè of congruences, where the b j are pairwise relatively prime, has as solution set èzè në x ç j=1 " cjë ëèa j, b j è+b j mëmodc where c j = C b j and C = b k. Moreover, the intersection contains only one element, i.e., one residue class. è
5 CHAPTER 2. REMODULIZATION OF CONGRUENCES 5 Proof. In order to show that this element exists, we consider the following: ç x ç a1 mod b1 x ç a2 mod b2 where 0 ç a1 é b1 and 0 ç a2 é b2 and gcdèb1;b2è = 1. Remodulizing the congruences by the factors b2 and b1, respectively, x ç ëa1;a1 + b1;::: ;a1 +èb2, 1è b1 ëmodb1b2 x ç ëa2;a2 + b2;::: ;a2 +èb1, 1è b2 ëmodb1b2 it is required to show that the sets of initial elements intersect, i.e., there exist integers k and h where 1 ç k ç b2, 1 and 1 ç h ç b1, 1, such that a1 + kb1 = a2 + hb2. Rewriting this equation, we require integers k and h such that kb1, hb2 = a2, a1. Since b1 and b2 are relatively prime, èa2, a1è is divisible by gcdèb1;b2è. Now notice that if k and h are solutions to kb1, hb2 = 1, then kèa2, a1è and hèa2, a1è are solutions to the required equation. Euclid's algorithm insures that such integers k and h exist. It follows that the ærst pair of congruences have a solution. We wishnowto show that the pair has a unique solution modulo b1b2. We know that a solution exists, that is, there exists an integer x such that x 2fa1;a1 + b1;::: ;a1 +èb2, 1èb1g ë fa2;a2 + b2;::: ;a2 +èb1, 1èb2g: Now suppose that the two initial sets intersect in two elements, say and x1 = a1 + çb1 = a2 + kb2 x2 = a1 + çb1 = a2 + hb2 : Subtracting the second formulation from the ærst for each x i, so that a1, a2 = kb2, çb1 = hb2, çb1 ; èk, hèb2 =èç, çèb1 : Since b1 and b2 are relatively prime it must be that k, h = mb1 and ç, ç = nb2 ; for some integers m and n. In other words, k = h + mb1 and ç = ç + nb2
6 CHAPTER 2. REMODULIZATION OF CONGRUENCES 6 so that x1 becomes This says that x1 = a1 +èç + nb2èb1 = a2 +èh + mb1èb2 = a1 + çb1 + nb2b1 = a2 + hb2 + mb1b2 : a1 + çb1 = a2 + hb2 +èm, nèb1b2 ; which isx2. This implies that m, n = 0 ; i.e., m = n, and hence Therefore, k = h + mb1 ç = ç + mb2 : x1 = a1 + çb1 = a1 +èç + mb2èb1 = a1 + çb1 + mb1b2 = x2 + mb1b2 : This means that x1 and x2 are in the same residue class mod b1b2; i.e., they are congruent mod b1b2 and the solution set is given as the unique class x ç d mod b1b2 ; where d ç x1 mod b1b2 ç x2 mod b1b2 from above. In the event there exist three congruences, we solve the ærst two congruences and combine this result with the third congruence, i.e., ç x ç d mod b1 b2 x ç a3 mod b3 and repeat the argument since b1b2 and b3 are relatively prime. The induction works and both the existence and the uniqueness are established. Suppose we want to solve cx ç a mod b for x, where 1 é c é b and a is divisible by gcdèc; bè èotherwise no solution existsè. We consider the case where gcdèc; bè = 1. By remodulizing amodbby the factor c, we obtain cx ç ëa; a + b;::: ;a+èc, 1è b ëmodcb: Since the set fa; a+b; : : : ; a+èc,1è bg forms a complete residue system mod c, there exists an element in this set, call it d, which is divisible by c. Since cx ç ëa; a + b;::: ;d;::: ;a+èc, 1èbë modcb we ænd that the only congruence solvable is cx ç d mod cb. The remaining congruences, cx ç ëa; a + b;::: ;d, b; d + b; : : : ; a +èc, 1è b ëmodcb
7 CHAPTER 2. REMODULIZATION OF CONGRUENCES 7 are not solvable, since in each case the factor c is pairwise relatively prime with the elements fa; a+b;::: ;d,b; d+b; : : : ; a+èc,1èbg, and thus does not divide them. In the congruence cx ç d mod cb, dividing through by c, x ç d c mod cb c or x ç d c mod b: Note that d c éb. To illustrate this procedure, consider the following example. Suppose 5x ç 3 mod 7. This is solvable since 3 is divisible by gcdè5; 7è=1. Remodulizing 3 mod 7by the factor 5 gives 5x ç ë3; 10; 17; 24; 31ë mod 5 æ 7 so that 5x ç 10 mod 35 is the only possible solution and, upon dividing all three terms by 5, x ç 2mod7: Note that 5x ç ë3; 17; 24; 31ë mod 35 does not yield any solutions, since in this case gcdè5; 35è = 5 does not divide any number in the set f3; 17; 24; 31g. The remodulization method also provides a way of ænding solutions to systems of congruences using linear congruences. Suppose we have the following system, ç x ç a1 mod b1 x ç a2 mod b2 where b1 éb2 and b1 and b2 are relatively prime. Theideaistomultiply the ærst congruence by b2 and the second congruence by b1, i.e., ç b2 x ç a1b2 mod b1b2 b1x ç a2b1 mod b1b2 so that we obtain a common modulus. By subtracting the second linear congruence from the ærst, we obtain a single linear congruence, èb2, b1èx ç èa1b2, a2b1è modb1b2 : The unique solution èmodulo b1b2è is insured, since gcdèb2, b1;b1b2è = 1. If the system consists of more than two congruences, then the solution of the ærst two congruences is combined with the third, and so on, to obtain a solution for the entire system.
8 CHAPTER 2. REMODULIZATION OF CONGRUENCES 8 Corollary 1 A system of linear congruences 8 é é: c1x ç a1 mod b1 c2x ç a2 mod b2. c n x ç a n mod b n where 1 éc j éb j,theb j are pairwise relatively prime, and the a j are divisible by gcdèc j ;b j è,can be reduced to a system 8 é é: x ç d1 mod b1 x ç d2 mod b2. x ç d n mod b n : By Theorem 2, the solution to this system is në x ç j=1 " cjë ëèd j, b j è+b j mëmodc where c j = 1 b j b k and C = b k, and the intersection contains only one residue class. è References ë1ë Burton, David M. Elementary Number Theory, Second Edition. Wm. C. Brown Publishers, Dubuque, Iowa, ë2ë Gold, J. F. and Don H. Tucker. Applications. To be submitted. Remodulization of Congruences and Its ë3ë Ore, Oystein. Number Theory and Its History. New York, Dover Publications, Inc., ë4ë Stewart, B. M. Theory of Numbers. The MacMillan Co., New York, 1952.
Chapter 4 Complementary Sets Of Systems Of Congruences Proceedings NCUR VII. è1993è, Vol. II, pp. 793í796. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationChapter 8 Vector Products Revisited: A New and Eæcient Method of Proving Vector Identities Proceedings NCUR X. è1996è, Vol. II, pp. 994í998 Jeærey F. Gold Department of Mathematics, Department of Physics
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationChapter 1 A Pri Characterization of T m e Pairs w in Proceedings NCUR V. (1991), Vol. I, pp. 362{366. Jerey F. Gold Department of Mathematics, Department of Physics University of Utah DonH.Tucker Department
More informationNumber Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures
Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it
More informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More informationComputer and Network Security
MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationHandout NUMBER THEORY
Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationKevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm
MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationAn Introductory Course in Elementary Number Theory. Wissam Raji
An Introductory Course in Elementary Number Theory Wissam Raji 2 Preface These notes serve as course notes for an undergraduate course in number theory. Most if not all universities worldwide offer introductory
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationTHE CONGRUENT NUMBER PROBLEM
THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationQUADRATIC RECIPROCITY IN CHARACTERISTIC 2
QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets
More informationGalois Theory, First Edition
Galois Theory, First Edition David A. Cox, published by John Wiley & Sons, 2004 Errata as of September 18, 2012 This errata sheet is organized by which printing of the book you have. The printing can be
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More informationOverview of Number Theory Basics. Divisibility
Overview of Number Theory Basics Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Divisibility Definition Given integers a and b, b 0, b divides a (denoted b a) if integer c, s.t. a = cb. b is called
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More information4. FIRST STEPS IN THE THEORY 4.1. A
4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We
More information51 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8
51 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8 Note: Integers are the w hole numbers and their negatives (additive inverses). While our text discusses only whole numbers, all these ideas
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationRESULTANT AND DISCRIMINANT OF POLYNOMIALS
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results
More informationcalculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,
Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationPrimality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationFactorization Algorithms for Polynomials over Finite Fields
Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 20110503 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationOn Generalized Fermat Numbers 3 2n +1
Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic
More informationFACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS
International Electronic Journal of Algebra Volume 6 (2009) 95106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationTEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011
TEXAS A&M UNIVERSITY Prime Factorization A History and Discussion Jason R. Prince April 4, 2011 Introduction In this paper we will discuss prime factorization, in particular we will look at some of the
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
More informationPolynomials and Factoring
Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationThe Factor Theorem and a corollary of the Fundamental Theorem of Algebra
Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationMaths delivers! A guide for teachers Years 11 and 12. RSA Encryption
1 Maths delivers! 2 3 4 5 6 7 8 9 10 11 12 A guide for teachers Years 11 and 12 RSA Encryption Maths delivers! RSA Encryption Dr Michael Evans AMSI Editor: Dr Jane Pitkethly, La Trobe University Illustrations
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationFACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set
FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationCryptography and Network Security Number Theory
Cryptography and Network Security Number Theory XiangYang Li Introduction to Number Theory Divisors b a if a=mb for an integer m b a and c b then c a b g and b h then b (mg+nh) for any int. m,n Prime
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationPrimality Testing and Factorization Methods
Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationPROOFS BY DESCENT KEITH CONRAD
PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More informationCIS 5371 Cryptography. 8. Encryption 
CIS 5371 Cryptography p y 8. Encryption  Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: Allornothing secrecy.
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationWOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology
First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More informationCryptography and Network Security Chapter 8
Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:
More informationOn the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 10361040, 2012 ISSN 15493636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
More informationAbstract Algebra Theory and Applications. Thomas W. Judson Stephen F. Austin State University
Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University August 16, 2013 ii Copyright 19972013 by Thomas W. Judson. Permission is granted to copy, distribute and/or
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B
More informationAMBIGUOUS CLASSES IN QUADRATIC FIELDS
MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES 0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the
More informationIntroduction to Modern Algebra
Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationON UNIQUE FACTORIZATION DOMAINS
ON UNIQUE FACTORIZATION DOMAINS JIM COYKENDALL AND WILLIAM W. SMITH Abstract. In this paper we attempt to generalize the notion of unique factorization domain in the spirit of halffactorial domain. It
More informationProbability Using Dice
Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More information