Elementary Number Theory


 Arline Davidson
 2 years ago
 Views:
Transcription
1 Elementary Number Theory Ahto Buldas December 3, 2016 Ahto Buldas Elementary Number Theory December 3, / 1
2 Division For any m > 0, we define Z m = {0, 1,... m 1} For any n, m Z (m > 0), there are unique q Z and r Z m such that: n = qm + r, where r is called the reminder (of n modulo m) and is denoted by r = n mod m. If r = 0, we say that m divides n (or n is divisible by m) and write m n. If 0 n < m, then r = n; if m n < 2m, then r = n m Z m, etc. If m n < 0, then r = n + m; if 2m n < m, then r = n + 2m, etc. Ahto Buldas Elementary Number Theory December 3, / 1
3 Equivalence of Numbers modulo m If a mod m = b mod m (i.e. if a b = km for a k Z, or m (a b)), then we write a b (mod m), and say that a and b are equivalent modulo m. For example 1 2 (mod 3), 7 1 (mod 3), 2 12 (mod 5), etc. Ahto Buldas Elementary Number Theory December 3, / 1
4 Z m as a Number Domain We can define addition and multiplication in Z m denoted by ja in the next way: For example, in Z 3 : and in Z 5 : a b = (a + b) mod m, a b = (a b) mod m. 2 2 = 2 2 = 1, 1 2 = 0, 2 3 = 0, 3 3 = 1 = 3 2 and 3 4 = 2. Ahto Buldas Elementary Number Theory December 3, / 1
5 Properties of the Function mod m: Z Z m mod m is a projector: (a mod m) mod m = a mod m. mod m preserves the operations (i.e. is a homomorphism): If a = a mod m, b = b mod m ja c = c mod m, then Conclusion 1: When computing a + b = c = a b = c a b = c = a b = c. a + b (c + d (e + f))... mod m we can reduce mod m whenever we want. Conclusion 2: and are somewhat similar to ordinary + and Ahto Buldas Elementary Number Theory December 3, / 1
6 Properties of the Z m Number Domain Though and differ from + and, we mostly use + and if this will not cause confusion. The following properties hold in Z m : Commutativity: a + b = b + a, a b = b a Associativity: (a + b) + c = a + (b + c), (a b) c = a (b c) Zero: a + 0 = 0 + a = a, a 0 = 0 a = 0 Unit: a 1 = 1 a = a Distributivity: (a + b) c = a c + b c, Ahto Buldas Elementary Number Theory December 3, / 1
7 Somewhat Unusual Properties of Z m The inverse a of an element a Z m is m a Z m, because: a + (m a) = m 0 (mod m). Zero divisors: the product of two nonzero elements can be zero. For example, in Z 6 : (mod 6). Not every element a has an inverse a 1 in Z m : a a 1 1. For example, zero divisors never have inverses. Ahto Buldas Elementary Number Theory December 3, / 1
8 Motivation from Cryptography In cryptography, the operations should be invertible, because any encrypted message should later be decrypted. Both mod addition and multiplication are extensively used in cryptography. Modular addition is invertible, i.e. a x = b is always solvable. Modular multiplication is not always invertible, i.e. a x = b can be unsolvable. For example, 2 x 5 (mod 6) is not solvable. The equation 2 x 5 (mod 7) is solvable: x = 6, because 2 6 = 12 5 (mod 7). Ahto Buldas Elementary Number Theory December 3, / 1
9 Greatest Common Divisor By the greatest common divisor gcd(a, b) of two nonnegative numbers a and b (not both zero!) we mean the largest d that divides both numbers, i.e.: gcd(a, b) = max{d: d a and d b}. Theorem An element a Z m is invertible if and only if gcd(a, m) = 1. Ahto Buldas Elementary Number Theory December 3, / 1
10 Computing gcd(a, b): Euclid s Algorithm For a > b 0: gcd(a, b) = { a if b = 0 gcd(b, a mod b) if b 0 (1) The work of Euclid s algorithm can be represented as a sequence: gcd(r 0, r 1 ) = gcd(r 1, r 2 ) =... = gcd(r m 1, r m ) = gcd(r m, 0), where r 0 = a, r 1 = b, and r k+1 = r k 1 mod r k < r k for any k > 1. This algorithm stops (an m with r m+1 = 0 exist), because otherwise r 0 > r 1 > r 2 >... > r k >... is an infinite decreasing sequence of natural numbers, which does not exist. Ahto Buldas Elementary Number Theory December 3, / 1
11 Correctness of Euclid s Algorithm Clearly gcd(a, 0) = a. We prove gcd(a, b) = gcd(b, a mod b), if a > b > 0. If D a,b = {d: d a and d b} is the set of all common divisors of a and b: gcd(a, b) = max D a,b and gcd(b, a mod b) = max D b,a mod b. It is sufficient to prove that D a,b = D b,a mod b. This is indeed the case, as: If d a ja d b, then d (a mod b) = a kb, and hence D a,b D b,a mod b If d (a mod b) and d b, then also d a, because a = (a mod b) + kb, and hence D a,b D b,a mod b. Ahto Buldas Elementary Number Theory December 3, / 1
12 Efficiency of Euclid s Algorithm Theorem Euclid s algorithm finds gcd(a, b) using 1.44 log 2 b + 1 divisions. Let r 0 > r 1 >... r n 1 > r n be the sequence produced by Euclid s algorithm so that r n = gcd(a, b). Let φ = , i.e. 1 + φ 1 = φ. We show by induction that r k φ n k for 1 k m, i.e. b = r 1 φ n 1. As r k+1 = r k 1 mod r k = r k 1 q k r k, we have r k 1 = q k r k + r k+1, where q k 1 because of r k 1 > r k. Basis: r n = gcd(a, b) 1 = φ 0. As r n+1 = 0 and q n r n = r n 1 > r n, we have q n 2 and hence r n 1 2 > φ 1. Step: If r k+1 φ n k 1 and r k φ n k, then r k 1 = q k r k +r k+1 r k +r k+1 = φ n k 1 +φ n k = φ n k (1+φ 1 ) = φ n k+1. Ahto Buldas Elementary Number Theory December 3, / 1
13 Conclusions Conclusion 1: If a > b 0, then there exist α, β Z such that gcd(a, b) = αa + βb. Conclusion 2: gcd(a, b) = 1 if and only if α, β Z, such that αa + βb = 1. Proof: If gcd(a, b) = 1, then use Conclusion 1. If α, β Z such that αa + βb = 1, (2) d a and d b, then d 1 by (??), i.e. gcd(a, b) = 1. Conclusion 3: If gcd(a, m) = 1, then b Z m, such that b a mod m = 1. Proof: Given α, β Z, so that αa + βm = 1, define b = α mod m. Ahto Buldas Elementary Number Theory December 3, / 1
14 Finding Inverses with Euclid s Algorithm Find 1 3 mod 26. Let a = 3 and b = a b 3 2 a b 8a 1 2 a (b 8a) = 9a b b 8a 1 0 9a b b 8a 2(9a b) = 26a + 3b Hence, = 1, which means (mod 26) Ahto Buldas Elementary Number Theory December 3, / 1
15 Solvability of ax mod n = c Theorem The equation ax mod n = c (where c Z n ) is solvable iff gcd(a, n) c. Proof. If the equation is solvable and d = gcd(a, n), then a, n, k Z so that a = a d, n = n d, and hence d c, because: c = ax mod n = ax kn = a dx kn d = (a x kn )d. If d = gcd(a, n) c, then gcd( a d, n d ) = 1, which means that a d has inverse modulo n d and the equation a d x mod n d = c d is solvable, i.e. k Z: a d x k n d = c d, and hence ax kn = c Z n, which means that ax mod n = c. Ahto Buldas Elementary Number Theory December 3, / 1
16 How Many Invertible Elements mod m are there? Answer to that question is called the Euler s function ϕ(m). Computing ϕ(m) requires the primefactorization of m. A prime number is a number if it has exactly two divisors. For example: 2, 3, 5, 7, 11, 13, etc. Theorem (Fundamental Theorem of Arithmetics) Every integer m > 0 has a unique prime factorization: p e 1 1 pe pe k k, where p 1 < p 2 <... < p k are prime numbers. For example: 60 = Ahto Buldas Elementary Number Theory December 3, / 1
17 Some Lemmas Lemma 1: Every composite m 2 is a product of primes. Proof: Let m be the smallest composite number that is not a product of primes. Hence, there exist composite numbers m 1, m 2 < m, so that m = m 1 m 2. Hence, m 1 and m 2 are products of primes and so must be m. A contradiction. Lemma 2: If gcd(a 1, b) = 1 = gcd(a 2, b), then gcd(a 1 a 2, b) = 1. Proof: As there are α 1, β 1, α 2, β 2, so that α 1 a 1 + β 1 b = 1 = α 2 a 2 + β 2 b: 1 = (α 1 a 1 + β 1 b) (α }{{} 2 a 2 + β 2 b) = α }{{} 1 α 2 a }{{} 1 a 2 + (β 1 + α 1 a 1 β 2 ) b, }{{} 1 1 α β we have gcd(a 1 a 2, b) = 1. Ahto Buldas Elementary Number Theory December 3, / 1
18 Fundamental Theorem of Arithmetics: Proof Theorem Every composite m 2 has a unique primefactorization p 1 p 2... p k, where p 1 p 2... p k. Proof. Let m be the smallest number that has two different primefactorisations: p 1 p 2... p k = m = q 1 q 2... q l. Hence, p i q j, because otherwise m = m/p i < m also has two different factorizations. Thus, gcd(p 1, q 1 ) = gcd(p 2, q 1 ) =... = gcd(p k, q 1 ) = 1, which by the assumption q 1 m and Lemma 2 implies a contradiction: q 1 = gcd(m, q 1 ) = gcd(p 1 p 2... p k, q 1 ) = 1. Ahto Buldas Elementary Number Theory December 3, / 1
19 Computing the Euler s Function Theorem If m = p e 1 1 pe pe k k ( ϕ(m) = = m p e 1 1 pe is the prime decomposition, then ) (1 1p1 ) ( p e 2 2 pe ) (1 1p2 )... (... ) p e k k p e k 1 k ) (1 1pk. The proof uses the inclusionexclusion principle from counting theory. Ahto Buldas Elementary Number Theory December 3, / 1
20 InclusionExclusion Principle Let P 1,..., P k be subsets of a set M. We want to count those elements of M that belong to none of P n, i.e. we want to compute M\ n P n. If k = 1, then M\ n P n = M P 1. If k = 2, then M\ n P n = M P 1 P 2 + P 1 P 2. If k = 3, then: M\ n P n = M P 1 P 2 P 3 + P 1 P 2 + P 1 P 3 + P 2 P 3 P 1 P 2 P 3. General case: M\ n P n = M Σ 1 + Σ 2 Σ ( 1) i Σ i where Σ i = (j 1,...,j i ) c(i) P j 1... P ji and the summation is over the set c(i) of all icombinations of indices 1, 2,..., k. There are ( k i) of them. Ahto Buldas Elementary Number Theory December 3, / 1
21 InclusionExclusion Principle and Euler s function Let M = Z m, where m = p e 1 1 pe pe k k. Let P n be the set of elements in Z m divisible by p n. Then ϕ(m) = M\ n P n This is because a Z m is invertible iff none of p 1,... p k divides a. P i = m p i, P i P j = m p i p j... P i1... P il = m p i1 p i2...p il. and hence: ϕ(m) = m m... m + m m m... p 1 p k p 1 p 2 p k 1 p k p 1 p 2 p ) ) ) 3 = m (1 1p1 (1 1p2... (1 1pk. Ahto Buldas Elementary Number Theory December 3, / 1
Elementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
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 informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
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 informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
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 informationDiscrete Mathematics, Chapter 4: Number Theory and Cryptography
Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationCLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010
CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationMathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR
Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGrawHill The McGrawHill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,
More informationSUM OF TWO SQUARES JAHNAVI BHASKAR
SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted
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 informationGREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationFinite Fields and ErrorCorrecting Codes
Lecture Notes in Mathematics Finite Fields and ErrorCorrecting Codes KarlGustav Andersson (Lund University) (version 1.01316 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents
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 information3. Applications of Number Theory
3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving NonConditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationMATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS
MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:003:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:
More informationIntegers and division
CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.
More informationGCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!
GCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!!! Challenge Problem 2 (Mastermind) due Fri. 9/26 Find a fourth guess whose scoring will allow you to determine the secret code (repetitions are
More informationWinter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com
Polynomials Alexander Remorov alexanderrem@gmail.com Warmup Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).
More informationAppendix A. Appendix. A.1 Algebra. Fields and Rings
Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.
More informationDefinition 1 Let a and b be positive integers. A linear combination of a and b is any number n = ax + by, (1) where x and y are whole numbers.
Greatest Common Divisors and Linear Combinations Let a and b be positive integers The greatest common divisor of a and b ( gcd(a, b) ) has a close and very useful connection to things called linear combinations
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
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 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 informationRSA and Primality Testing
and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2
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 informationElementary Number Theory
Elementary Number Theory A revision by Jim Hefferon, St Michael s College, 2003Dec of notes by W. Edwin Clark, University of South Florida, 2002Dec L A TEX source compiled on January 5, 2004 by Jim Hefferon,
More informationGroups in Cryptography
Groups in Cryptography Çetin Kaya Koç http://cs.ucsb.edu/~koc/cs178 koc@cs.ucsb.edu Koç (http://cs.ucsb.edu/~koc) ucsb cs 178 intro to crypto winter 2013 1 / 13 Groups in Cryptography A set S and a binary
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 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 informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More information26 Ideals and Quotient Rings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed
More informationCOMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
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 informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationa = bq + r where 0 r < b.
Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationLectures on Number Theory. LarsÅke Lindahl
Lectures on Number Theory LarsÅke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder
More information15 Prime and Composite Numbers
15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such
More informationProof of Euler s φ (Phi) Function Formula
Rose Hulman Undergraduate Mathematics Journal Proof of Euler s φ (Phi Function Formula Shashank Chorge a Juan Vargas b Volume 14, no. 2, Fall 2013 Sponsored by RoseHulman Institute of Technology Department
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
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 informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
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 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 informationEvery Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
More informationIntegers, Division, and Divisibility
Number Theory Notes (v. 4.23 October 31, 2002) 1 Number Theory is the branch of mathematics that deals with integers and their properties, especially properties relating to arithmetic operations like addition,
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 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 informationTest1. Due Friday, March 13, 2015.
1 Abstract Algebra Professor M. Zuker Test1. Due Friday, March 13, 2015. 1. Euclidean algorithm and related. (a) Suppose that a and b are two positive integers and that gcd(a, b) = d. Find all solutions
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 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 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 informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationThe cyclotomic polynomials
The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationDIVISIBILITY AND GREATEST COMMON DIVISORS
DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationConsequently, for the remainder of this discussion we will assume that a is a quadratic residue mod p.
Computing square roots mod p We now have very effective ways to determine whether the quadratic congruence x a (mod p), p an odd prime, is solvable. What we need to complete this discussion is an effective
More informationPYTHAGOREAN TRIPLES PETE L. CLARK
PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =
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 informationSettling a Question about Pythagorean Triples
Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands EMail address:
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationLecture 13: Factoring Integers
CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method
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 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 information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBINCAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationBasic Algorithms In Computer Algebra
Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,
More informationNumber Theory: A Mathemythical Approach. Student Resources. Printed Version
Number Theory: A Mathemythical Approach Student Resources Printed Version ii Contents 1 Appendix 1 2 Hints to Problems 3 Chapter 1 Hints......................................... 3 Chapter 2 Hints.........................................
More information54 Prime and Composite Numbers
54 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
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 informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
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 informationPublic Key Cryptography: RSA and Lots of Number Theory
Public Key Cryptography: RSA and Lots of Number Theory Public vs. PrivateKey Cryptography We have just discussed traditional symmetric cryptography: Uses a single key shared between sender and receiver
More information