CHAPTER 5. Number Theory. 1. Integers and Division. Discussion


 Owen Hall
 5 years ago
 Views:
Transcription
1 CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 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 b. If a does not divide b, we write a b. Example The number 6 is divisible by 3, 3 6, since 6 = 3 2. Exercise Let a, b, and c be integers with a 0. Prove that if ab ac, then b c. Using this definition, we may define an integer to be even if it is divisible by 2 and odd if it is not divisible by 2. This concept is one of the simplest of properties of numbers to define, yet it is among the most complicated of all mathematical ideas. Keep in mind that we are talking about a very restricted notion of what it means for one number to divide another: we can certainly divide 7 by 3 and get the rational number 7 = , but, since the result is not an integer, we say that 3 does not 3 divide 7, or 3 7. For this reason, you should avoid using fractions in any discussion of integers and integer arithmetic Basic Properties of Divisibility. Theorem For all integers a, b, and c, 1. If a b and a c, then a (b + c). 2. If a b, then a (bc). 3. If a b and b c, then a c. 146
2 1. INTEGERS AND DIVISION 147 Theorem states the most basic properties of division. Here is the proof of part 3: Proof of part 3. Assume a, b, and c are integers such that a b and b c. Then by definition, there must be integers m and n such that b = am and c = bn. Thus c = bn = (am)n = a(mn). Since the product of two integers is again an integer, we have a c. Exercise Prove part 1 of Theorem Exercise Prove part 2 of Theorem Theorem The Division Algorithm. Theorem (Division Algorithm) Given integers a and d, with d > 0, there exists unique integers q and r, with 0 r < d, such that a = qd + r. Notation We call a the dividend, d the divisor, q the quotient, and r the remainder. The division algorithm is probably one of the first concepts you learned relative to the operation of division. It is not actually an algorithm, but this is this theorem s traditional name. For example, if we divide 26 by 3, then we get a quotient of 8 and remainder or 2. This can be expressed 26 = It is a little trickier to see what q and r should be if a < 0. For example, if we divide 26 is by 3, then the remainder is not 2. We can, however, use the equation 26 = to our advantage: 26 = 3 ( 8) 2 = [3 ( 8) 3] = 3( 9) + 1 So dividing 26 by 3 gives a quotient of 9 and remainder 1. The condition 0 r < d makes r and q unique for any given a and d Proof of Division Algorithm. Proof. Suppose a and d are integers, and d > 0. We will use the wellordering principle to obtain the quotient q and remainder r. Since we can take q = a if d = 1, we shall assume that d > 1. Let S be the set of all natural numbers of the form a kd, where k is an integer. In symbols S = {a kd k Z and a kd 0}. If we can show that S is nonempty, then the wellordering principle will give us a least element of S, and this will be the remainder r we are looking for. There are two cases.
3 1. INTEGERS AND DIVISION 148 Case 1: a 0. In this case, we can set k = 0 and get the element a 0 d = a 0 of S. Case 2: a < 0. In this case, we can set k = a. Then a kd = a ad = a(1 d). Since a < 0 and d > 1, a(1 d) > 0; hence is an element of S. Thus, S, and so S has a least element r = a qd for some integer q. Thus, a = qd + r and r 0. We are left to show (i) r < d and (ii) q and r are unique. (i) Suppose r d. Then r = d + r, where 0 r < r. Then a = qd + r = qd + d + r = (q + 1)d + r, so that r = a (q + 1)d is an element of S smaller than r. This contradicts the fact that r is the least element of S. Thus, r < d. (ii) Suppose integers q and r satisfy a = q d + r and 0 r < d. Without loss of generality, we may assume r r, so that 0 r r < d. Since a = q d + r = qd + r, r r = d(q q). This means that d divides r r, which implies either r r d or r r = 0. But but we know 0 r r < d. Thus, r = r, which, in turn, implies q = q. That is, q and r are unique Prime Numbers, Composites. Definition If p is an integer greater than 1, then p is a prime number if the only divisors of p are 1 and p. Definition A positive integer greater than 1 that is not a prime number is called composite. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or find its prime factors. This fact is the basis behind many of the cryptosystems currently in use. One problem is that there is no known procedure that will generate prime numbers, even recursively. In fact, there are many things about prime numbers that we don t know. For example, there is a conjecture, known as Goldbach s Conjecture, that there are infinitely many prime pairs, that is, consecutive odd prime numbers, such as 5 and 7, or 41 and 43, which no one so far has been able to prove or disprove. As the next theorem illustrates, it is possible, however, to prove that there are infinitely many prime numbers. Its proof, attributed to Euclid, is one of the most elegant in all of mathematics. Theorem There are infinitely many prime numbers.
4 1. INTEGERS AND DIVISION 149 Proof. We prove the theorem by contradiction. Suppose there are only finitely many prime numbers, say, p 1, p 2,..., p n. Let N = p 1 p 2 p n + 1. Then N is an integer greater than each of p 1, p 2,..., p n, so N cannot be prime. In Example 9, Module 3.3, we showed that N can be written as a product of prime numbers; hence, some prime p divides N. We may assume, by reordering p 1, p 2,..., p n, if necessary, that p = p 1. Thus N = p 1 a for some integer a. Substituting, we get p 1 a = p 1 p 2 p n + 1 p 1 a p 1 p 2 p n = 1 p 1 (a p 2 p n ) = 1. Thus, a p 2 p n is a positive integer. Since p 1 is a prime number, p 1 > 1, and so But this contradicts the equality above. p 1 (a p 2 p n ) > Fundamental Theorem of Arithmetic. Theorem (Fundamental Theorem of Arithmetic) Every positive integer greater than one can be written uniquely as a product of primes, where the prime factors are written in nondecreasing order. We have already given part of the proof Theorem in an example of Module 3.3 Induction. There we showed that every positive integer greater than 1 can be written as a product of prime numbers. The uniqueness of the factors is important, and the proof that they are unique, which requires a few additional ideas, will be postponed until the next module. The prime factorization of 140 is You can see one reason why we do not want 1 to be prime: There is no limit to the number of times 1 may be repeated as a factor, and that would give us nonunique prime factorizations Factoring. Theorem If n is a composite integer, then n has a factor less than or equal to n.
5 1. INTEGERS AND DIVISION 150 Theorem can be helpful in narrowing down the list of possible prime factors of a number. It was proved in an example of Module 3.2 Methods of Proof and exploited in another example of that module. If the number 253 is composite, for example, it must have a factor less than or equal to 15. Thus we need only check the primes 2, 3, 5, 7, 11, and 13. It turns out 253 = Mersenne Primes. Definition A prime number of the form 2 p 1, where p is a prime number, is called a Mersenne prime. Mersenne primes are a special class of primes, which lend themselves to a nice theoretical development. Not all primes are Mersenne, though, and not all numbers of the form 2 p 1 are prime. For example, 2 p 1 is prime for p = 2, 3, 5, and 7, but = 2047 = 23 89, which is not prime. On the other hand, the primes 5 and 11 cannot be written in this form Greatest Common Divisor and Least Common Multiple. Definitions Given integers a and b (1) The greatest common divisor of a and b, denoted GCD (a, b), is the largest positive integer d such that d a and d b. (2) The least common multiple of a and b, denoted LCM (a, b), is the smallest positive integer m such that a m and b m. (3) a and b are called relatively prime if GCD (a, b) = 1. (4) The integers a 1, a 2, a 3,..., a n are called pairwise relatively prime if GCD(a i, a j ) = 1 for 1 i < j n. (5) The Euler φ function is the function φ : Z + N defined by φ(n) = the number of positive integers less than n that are relatively prime to n. Lemma Suppose a and b are integers and m = LCM(a, b). If c is a positive integer such that a c and b c, then m c. Proof. Suppose a c and b c, but m c. By the division algorithm there are (unique) positive integers q and r such that c = mq + r and 0 r < m. Since m c, r 0; that is, r > 0. Write r = c mq. Since a and b both divide c and m, a and b both divide r. But this contradicts the fact that m is supposed to be the least positive integer with this property. Thus m c.
6 Theorem ab = GCD(a, b) LCM(a, b). 1. INTEGERS AND DIVISION 151 The proof of Theorem will be discussed in the next module. Example Here are some examples to illustrate the definitions above. (1) GCD(45, 60) = 15, since 45 = 15 3 and 60 = 15 4 and 15 is the largest number that divides both 45 and 60. (2) LCM(45, 60) = 180, since 180 = 45 4 = 60 3 and 180 is the smallest number that both 45 and 60 divide. (3) 45 and 60 are not relatively prime. (4) 45 and 16 are relatively prime since GCD(45, 16) = 1. (5) 4, 7, 13 and 55 are pairwise relatively prime. (6) φ(15) = 8 If we are given the prime factorizations of two integers, then it is easy to find their GCD and LCM. For example, 600 = and 220 = has greatest common divisor = 20 and least common multiple = Since prime factorizations can be difficult to find, however, this idea does not lead to an efficient way to compute GCD s. We will introduce an efficient algorithm in the next module that does not involve knowledge about prime factorizations. Exercise Let F (n) denote the nth term of the Fibonacci Sequence. Prove using induction that GCD(F (n), F (n 1)) = 1 for all integers n Modular Arithmetic. Definition Given integers a and m, with m > 0, a mod m is defined to be the remainder when a is divided by m. Definition a b(mod m), read a is congruent to b modulo (or mod) m, if m (a b); that is, (a b) mod m = 0. Theorem Given integers a, b, and m, 1. a b (mod m) if and only if a mod m = b mod m. 2. a b (mod m) if and only if a = b + km for some integer k. 3. If a b (mod m) and c d (mod m) then (a) a + c b + d (mod m) (b) a c b d (mod m)
7 1. INTEGERS AND DIVISION 152 The mod operation is derived from the Division Algorithm: If we divide the integer a by the positive integer m, we get a unique quotient q and remainder r satisfying a = mq + r and 0 r < m. The remainder r is defined to be the value of a mod m. One of the notational aspects that may seem a little unusual is that we write a + b(mod m) for (a + b)(mod m). Also, the symbol (mod m) may occasionally be omitted when it is understood. Example Here are some examples. (a) 12 mod 5 = 2 (b) 139 mod 5 = 4 (c) 1142 mod 5 = 2 (d) (mod 5) (e) (mod 5) (f) (mod 5) One of the differences to note between the concept of congruence modulo m verses the mod operator is that an integer, k may be congruent to infinitely many other integers modulo m, however, k mod m is equal to one single integer. For example, 139 mod 5 = 4, but 139 is congruent to all the elements of {..., 6, 1, 4, 9, 14, 19,... }. Exercise Given a positive integer m, prove that the assignment a a mod m defines a function f : Z Z. Is f onetoone? onto? What is its range? a a mod m is another way to write f(a) = a mod m. Here is a proof of part 3b of Theorem : Proof of 3b. Since a b(mod m) and c d(mod m), there must be integers s and t such that b = a + sm and d = c + tm (part 2). Thus bd = (a + sm)(c + tm) = ac + atm + smc + stm 2 = ac + (at + sc + stm)m Since a, c, t, and s are all integers, at + sc + st is as well. Thus, by part 2, ac bd(mod m). Exercise Prove part 3a of Theorem
8 1.11. Applications of Modular Arithmetic. 1. Hashing Functions 2. Pseudorandom Number Generators 3. Cryptology 1. INTEGERS AND DIVISION 153 There are many applications of modular arithmetic in computer science. One such application is in the construction of pseudorandom number generators. Numbers that seem to be somewhat random may be produced using the linear congruential method. As you will see, it does not produce truly random numbers, but rather a sequence of numbers that will eventually repeat. To generate a sequence we choose a modulus m, multiplier a, and an increment c. Then we start with a seed number x 0 and then construct a sequence of numbers recursively using the formula x n+1 = (ax n + c) mod m. Example Suppose we choose m = 11, a = 7, c = 3, and x 0 = 1. Then we get x 0 = 1 x 1 = ( )mod 11 = 10 x 2 = ( )mod 11 = 7 x 3 = ( )mod 11 = 8 x 4 = ( )mod 11 = 4 x 5 = ( )mod 11 = 9 x 6 = ( )mod 11 = 0 x 7 = ( )mod 11 = 3 etc. The sequence will be 1, 10, 7, 8, 4, 9, 0, 3, etc. If we wanted a random sequence of bits, 0 and 1, we could then reduce each x n mod 2. In practice, large Mersenne primes are often chosen for the modulus, and the repetition period for such sequences can be made to be quite large.
9 1. INTEGERS AND DIVISION 154 Exercise Prove that for a given modulus m, and arbitrary multiplier a, increment c, and seed x 0, the sequence x 0, x 1, x 2,... must eventually repeat.
8 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
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 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 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 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 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 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 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 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 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 informationMath 453: Elementary Number Theory Definitions and Theorems
Math 453: Elementary Number Theory Definitions and Theorems (Class Notes, Spring 2011 A.J. Hildebrand) Version 542011 Contents About these notes 3 1 Divisibility and Factorization 4 1.1 Divisibility.......................................
More informationV55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography
V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical
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 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 informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2  MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
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 informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
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 informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
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 informationMATH 13150: Freshman Seminar Unit 10
MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers
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 information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
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 informationTheorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime
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 informationSTUDENT S SOLUTIONS MANUAL ELEMENTARY NUMBER THEORY. Bart Goddard. Kenneth H. Rosen AND ITS APPLICATIONS FIFTH EDITION. to accompany.
STUDENT S SOLUTIONS MANUAL to accompany ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS FIFTH EDITION Bart Goddard Kenneth H. Rosen AT&T Labs Reproduced by Pearson AddisonWesley from electronic files supplied
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 informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationStupid Divisibility Tricks
Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
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 informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
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 informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
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 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 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 informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationThe Euclidean Algorithm
The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have
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 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 informationPage 331, 38.4 Suppose a is a positive integer and p is a prime. Prove that p a if and only if the prime factorization of a contains p.
Page 331, 38.2 Assignment #11 Solutions Factor the following positive integers into primes. a. 25 = 5 2. b. 4200 = 2 3 3 5 2 7. c. 10 10 = 2 10 5 10. d. 19 = 19. e. 1 = 1. Page 331, 38.4 Suppose a is a
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 informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
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 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 informationMATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003
MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationContinued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
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 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 informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15yearold why
More informationGrade 6 Math Circles March 10/11, 2015 Prime Time Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going
More informationDiscrete Structures. Lecture Notes
Discrete Structures Lecture Notes Vladlen Koltun 1 Winter 2008 1 Computer Science Department, 353 Serra Mall, Gates 374, Stanford University, Stanford, CA 94305, USA; vladlen@stanford.edu. Contents 1
More informationA Course on Number Theory. Peter J. Cameron
A Course on Number Theory Peter J. Cameron ii Preface These are the notes of the course MTH6128, Number Theory, which I taught at Queen Mary, University of London, in the spring semester of 2009. There
More information17 Greatest Common Factors and Least Common Multiples
17 Greatest Common Factors and Least Common Multiples Consider the following concrete problem: An architect is designing an elegant display room for art museum. One wall is to be covered with large square
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 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 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 informationThe Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic 1 Introduction: Why this theorem? Why this proof? One of the purposes of this course 1 is to train you in the methods mathematicians use to prove mathematical statements,
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 information4.2 Euclid s Classification of Pythagorean Triples
178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
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 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 informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a longwinded way of asking for the value of 6 2000 mod
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
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 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 information