The Fundamental Theorem of Arithmetic


 Deborah Spencer
 1 years ago
 Views:
Transcription
1 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, the proof techniques you have been studying learning to use are a large part of your training. Unfortunately, the examples you have so far encountered are toy examples, easy to deal with pretty much devoid of interest. In class, I compared the toy examples to finger exercises for the piano, boring in of themselves but necessary first steps on the road to much more interesting things. The analogy between finger exercises proof exercises is not perfect, however. A beginning piano student will already be familiar with real piano music, so the point of the finger exercises is clear to him or her. By contrast, it is difficult to expose beginning math students to real proofs of real theorems; as a consequence, many beginning math students may fail to see the point of what they are learning. This hout, which presents a proof of the the Fundamental Theorem of Arithmetic (or FTA, is intended to address this problem to some extent by introducing some real mathematics that beginning math students are already in a position to appreciate. The FTA, even though is is familiar to gradeschoolers, is actually by no means an obvious result: every proof of it is hard, as mentioned in class there are even mathematical environments in which the (suitably adapted FTA turns out to be false. 2 The FTA is a powerful theorem with many useful consequences; this hout will discuss one or two of them. The proof presented here is a very good example of a proof by contradiction. It is clever delicate in structure, it could even be termed an aesthetically pleasing wor of art. 3 A close reading of the proof will strengthen your ability to follow involved logical arguments to verify their correctness. Some of the exercises focus on subtle points in the argument. 2 The Theorem the Proof. 2.1 What the FTA says. In class, I mentioned the fact any integer n 2 has the following property: Unique factorization property for integers n 2: If two people factor n 2 into prime factors, they must get the same result; that is, if they both arrange the factors they get in nondecreasing order, the two sequences will match exactly. This fact is exactly what the FTA claims. Some notation will help me both to state the theorem precisely to explain the proof. 1 The other purpose is to introduce some basic, ubiquitous ideas, mostly from set theory. 2 The example mentioned in class turned on the equation (1 + i 5 (1 i 5 = 6 = I hasten to add that this proof is not original with me. I came across it in What is Mathematics? by Courant Robbins. 1
2 Let us say that any integer n 2 that has the unique factorization property above is a ufp integer. 4 Theorem 1 (The Fundamental Theorem of Arithmetic, or FTA. Every integer n 2 is a ufp integer. 2.2 A proof of the FTC. As indicated above, I will give a proof by contradiction. In order to mae the argument as clear as possible, I will separate out some pieces as claims; some small pieces will be left as exercises. Suppose, in order to get a contradiction, that Theorem 1 were 5 false. Then there would exist at least one integer n 2 that does not have the unique factorization property. Now, if you test all of the integers in order first 2, then 3, then 5, so on to see whether or not each one is a ufp integer, you would in this case eventually arrive at the first nonufp integer n 0 ; since n 0 would be the first nonufp integer you had found, it would also clearly be the smallest nonufp integer. Furthermore, since n 0 would be a nonufp integer, it would possess two nonidentical prime factorizations n 0 = p 1 p i p r n 0 = q 1 q j q s, where p 1 p i p r q 1 q j q s. The first step in the proof is to establish that the two factorizations in (1 would not only be nonidendical but in fact would have to be completely different, with no factors in common whatsoever: Claim 1.1 In (1, for every 1 i r 1 j s, p i q j ; that is, no p would be equal to any q. Proof (Claim 1.1. Observe first, that by (1, p 1 p r = q 1 q s. (2 Now if, say, p i0 = q j0, we could cancel it from both sides of (2 to get the equality (1 n 1 = p 1 p r p i0 = q 1 q s q j0. (3 Now, as is easy to show, 2 n 1 < n 0 (see Exercise 4. Since 2 n 1, (3 would give two different prime factorizations of n 1. On the other h, all integers in the interval [ 2, n 0 would be ufp integers, so that n 1 would not have two different prime factorizatioms. Since we have arrived at a contradiction, we can conclude that none of the p s could equal any of the q s. (Claim 1.1 We need one particular consequence of Claim 1.1, namely that p 1 q 1 could not be the same prime; 6 without loss of generality, let us say that that p 1 > q 1. Since p 1 > q 1, we can use the two factorizations in (1 to define the positive integer 7 x := ( p 1 p 2 p r ( q1 p 2 p r = (p1 q 1 p 2 p r (4 to ascertain some properties of x. The first property concerns how big or small x could be. 4 Of course, as the FTP says, all integers n 2 are ufp integers, but we cannot assume this fact before we prove it,, until we prove it, this special name will be useful. 5 I will eep the verbs contrarytofact to remind you that none of this is actually the case. Proofs by contradiction are not in general written this fastidiously! 6 This is the only consequence that it is safe to use at this point: we do not necessarily have more than one pprime that is, we cannot rule out the possibility that r = 1 in (2. 7 This definition is the important idea at the center of this proof! 2
3 Claim 1.2 The integer x would have to satisfy the inequalities 2 x < n 0. (5 Proof (Claim 1.2. It follows easily from (4 that x would have to be less than n 0 (see Exercise (1. To prove the assertion that x 2, I will consider two cases. Case 1. If p 1 = 3 in (1, so that q 1 = 2 (why??, then in the factorization n 0 = p 1 p i p r, (6 the index r would need to be at least 2 (see Exercise 2; therefore, starting from Equation (4, we can calculate: x = (p 1 q 1 p 2 p r = (3 2p 2 p r = p 2 p r (because p 2 is really there 2. Case 2. If, on the other h, p 1 3 in (1, then necessarily (p 1 q 1 2 (see Exercise 3, so that x = (p 1 q 1 p 2 p r (2p 2 p r (even if (p 2, p 3,..., p r are not there 2. (Claim 1.2 Claim 1.3 The integer x would have to be a multiple of q 1 ; that is, Proof (Claim 1.3. From (4, we have that x would be the difference q 1 x. (7 x := (p 1 p 2 p r (q 1 p 2 p r. (8 } {{ } } {{ } (α (β Now, the integer (β above is already expressed as a multiple of q 1, while the integer (α by (1 would be n 0, which also would have factorization q 1 q s. Thus, (8 expresses x as the difference of two multiples of q 1 ; this guarantees that x would itself be a multiple of q 1. (Claim 1.3 I am now in a position to assemble the pieces finish the proof. Claim 1.3 says that q 1 x, so that x = q 1 l (9 for some integer l, (9 would be the beginning of a prime factorization of x. On the other h, the right side of (4 would also be the beginning of a prime factorization of x, since 2 x < n 0 (this is Claim 1.2, x would be a ufp integer so that these two factorizations would have to match. In particular: since q 1 : would appear in the prime factorization that started with (9, q 1 would appear in the prime factorization of (p 1 q 1 p 2 p r. (10 Furthermore, since q 1 p i for 2 i r, q 1 would have to show up in the prime factorization of the (p 1 q 1 part of (10. In other words, (p 1 q 1 = q 1 (other primes, 3
4 or, more simply, Finally, adding q 1 to both sides of (11 gives the equation (p 1 q 1 = q 1 (. (11 p 1 = q 1 ( + 1, which would display the prime number p 1 as a product of two integers each of which was 2, contradicting the assumption that p 1 is prime. Exercise 1 Use (4 to show that x would have to be strictly less than n 0. Exercise 2 In (6: show that if p 1 = 3, then r could not equal 1. Exercise 3 Let p 1 q 1 be primes, with p 1 > q 1 p 1 3. Show that (p 1 q 1 2. Exercise 4 Let n 1 be the integer defined in Equation (3. [a]: Show that n 1 < n 0. [b]: Show that 2 n 1. 3 Some Applications. Theorem 1 is called The Fundamental Theorem of Arithmetic because so many of the properties of N Z depend upon it. In this hout, I will discuss a few of the most accessible examples. 3.1 Prime factorizations written using prime powers. Many (most? of the applications of the FTA are most conveniently stated by expressing prime factorizations with multiple copies of the same prime gathered together. For example, instead of writing the prime factorization of 504 as 504 = , one generally writes 504 = or 504 = (12 For some purposes, the form on the left side of (12 is more convenient, while for other applications, 8 the form on the right side of (12 is essential. When two or more integers are being considered at one time, in order to facilitate comparisons among them, it is customary to write each prime factorization so as to have it include every prime that appears in any of the factorizations. For example, if the numbers being considered are , one would generally write 8 Including the famous socalled Gödel numbering. 189 = = , or 189 = =
5 3.2 Divisibility; the gcd; the lcm. The two most basic consequences of the FTA are contained in Exercise 5. Exercise 5 Let n = p a1 1 pa2 2 par r let = p b1 1 pb2 2 pbr r be integers 9 2 with the prime factorizations shown. [a]: Show that [b]: Show that n = p a1+b1 1 p a2+b2 2 p ar+br r. n a 1 b 1 a 2 b 2. a r b r Two further consequences of the FTA concern least common multiples greatest common divisors: Definition 1 Let n be positive integers. [a]: The greatest common divisor of n, denoted gcd(n,, is the largest integer that divides both n. [b]: If gcd(n, = 1, n are said to be relatively prime. [c]: The least common multiple of n, denoted lcm(n,, is the smallest integer that is a multiple of both n. Obviously, for any integer n 1, gcd(1, n = 1 lcm(1, n = n. If n 2 2, Theorem 2 shows how to use the prime factorizations of n 10 to find gcd(n, lcm(n,. In order to state Theorem 2, I find the following notation to be very helpful: for integers 11 a b, let max(a, b be the larger of the two integers let min(a, b be the smaller of the two integers. 12 Theorem 2 Let n = p a1 1 pa2 2 par r Then Proof of Equation (13. Let us put let = p b1 1 pb2 2 pbr r be integers 2 with the prime factorizations shown. lcm(n, = p max(a1,b1 1 p max(a2,b2 2 p max(ar,br r (13 gcd(n, = p min(a1,b1 1 p min(a2,b2 2 p min(ar,br r. (14 M := p max(a1,b1 1 p max(a2,b2 2 p max(ar,br r. To show that M = lcm(n,, I must show that M satisfies the conditions of Definition 1[c]. Step I: Show that n M M. Since a i max(a i, b i for all 1 i r, we can apply Exercise 5[b] to conclude that n M. Similarly, since b i max(a i, b i for all 1 r, we can apply Exercise 5[b] to conclude that M. 9 if one of the numbers equals 1 (say n = 1, the exercise still maes sense is still true. You just let a 1 = = a r = 0 in that case. 10 if the prime factorizations are available. For large integers, prime factorizations are very hard to find, but there is an efficient way to find gcd(n, directly, bypassing prime factorization. 11 They need not be integers; this notation maes sense for all real numbers. However, in this hout, I will be using it only for integers. 12 For the case a = b, obviously, max(a, a = min(a, a = a. 5
6 Step II: Show that M is the smallest among all the common multiples of n. I will actually prove a much stronger statement: I will show that every common multiple of n is a multiple of M. The argument runs as follows. Let C be any common multiple of n. Then, by Exercise 5[b], the prime factorization of C can be written C = p c1 1 pc2 2 pcr r q d1 1 qds s, where the q d1 1 qds s is there to represent whatever primes there are that divide C but do not divide a or b. Now, again by Exercise 5[b], n C = ai c i (1 i r, C = bi c i (1 i r. But then (for each 1 i r, since both a i c i a i c i, obviously max(a i, b i c i. With a final application of Exercise 5[b], we can then conclude that M C. (Equation (13 The proof of Equation (14 is left as Exercise 6. (Theorem 2 Exercise 6 Prove Equation (14. Exercise 7 Use Theorem 2 to prove, for any positive integers n, that 3.3 Recovering bypassed results. gcd(, n lcm(, n = n. As I mentioned in class: the usual way to prove the FTA is to use some preliminary results, which will have been established beforeh. This hout, though, begins by establishing the FTA. We therefore need to derive the results that we have bypassed they are important in their own right. Exercise 8 ass you to derive two of these. Exercise 8 Use the unique factorization property to prove the following important facts about N. [a]: Let n be relatively prime. Show, for any integer l, that if n l then n l. [b]: Let p be prime, let n be integers. Show that if p n, then either p n or p (or both. 3.4 Square roots of positive integers. In class ( in the homewor, we showed that 2 6 are not rational, but we did not answer the full question, Exactly which positive integers have rational square roots? With the FTA, we can now state prove the complete answer this question. I will brea off as a lemma the part of the proof that employs the FTA. 2. Lemma 3 Let n be positive integers. If is a nonwholenumber fraction, then so is Proof: Since is a nonwholenumber fraction, n is not a multiple of. Therefore, if their prime factorizations are given by n = p a1 1 pa2 2 par r = p b1 1 pb2 2 pbr r, (15 6
7 2: then necessarily b i > a i for at least one 1 i r. Let us suppose that b i0 > a i0. Consider 2 = n2 2 (Why?? = p2a1 1 p 2a2 2 p 2ai 0 i 0 p 2ar r p 2b1 1 p2b2 2 p 2bi 0 i 0 p 2br r 2 furthemore, since b i0 > a i0, obviously 2b i0 > 2a i0. Thus, n 2 is not a multiple of 2, so is not a whole number. Exercise 9 In the proof of Lemma 3: by passing to (15, I was tacitly assuming that n 2 2, this need not be the case. Plug this logical gap: consider the cases n = 1 = 1. Exercise 10 Prove that Lemma 3 remains true even if we drop the assumption that n are positive. Hint: Don t REPROVE Lemma 3; USE it! Theorem 4 A positive integer n has a rational square root if only if n is the square of an integer. Proof ( =: This is completely obvious: if n = 2 for some integer, then obviously n =, a rational number. Proof (= : I will prove the contrapositive statement: If n is not the square of an integer, then n is not rational. Proof of contrapositive: For any positive integer n, there are three logical possibilities for n: (a: n is an integer; (b: n is a nonwholenumber fraction; or (c: n is an irrational number. Now, the we are given that n is not the square of an integer, so possibility (a is ruled out; Lemma 3 rules out possibility (b. Thus, possibility (c is the correct one: n is an irrational number. Exercise 11 In the proof of Theorem 4: how exactly does Lemma 3 rule out possibility (b? ; 7
Math 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 informationPrime Numbers. Chapter Primes and Composites
Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are
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 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 information2 The Euclidean algorithm
2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In
More informationElementary 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 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 information2.1.1 Examples of Sets and their Elements
Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define
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 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 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 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 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 informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
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 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 informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationContinued fractions and good approximations.
Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our
More informationLecture 1: Elementary Number Theory
Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers
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 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 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 informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
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 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 informationCourse notes on Number Theory
Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
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 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 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 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 informationTheorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.
Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,
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 information31 is a prime number is a mathematical statement (which happens to be true).
Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to
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 informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationProof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232  Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the
More informationStudents in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are
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 information9. The Pails of Water Problem
9. The Pails of Water Problem You have a 5 and a 7 quart pail. How can you measure exactly 1 quart of water, by pouring water back and forth between the two pails? You are allowed to fill and empty each
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 informationAlgebra for Digital Communication
EPFL  Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationIntroduction to mathematical arguments
Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
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 informationDISCRETE MATHEMATICS W W L CHEN
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
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 informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationThis section demonstrates some different techniques of proving some general statements.
Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More informationChapter 6. Number Theory. 6.1 The Division Algorithm
Chapter 6 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
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 informationPractice Problems for First Test
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
More informationMathematical induction & Recursion
CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,
More informationChapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1
Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes
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
Mathematical S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! Mathematical S 0 S 1 S 2 S 3 S4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! S 4 Mathematical S 0 S 1 S 2 S 3 S5 S 6 S 7 S 8 S 9 S 10
More informationThe Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
More informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
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 informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical
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 informationFurther linear algebra. Chapter I. Integers.
Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides
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 informationMath 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (
More informationStanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.
Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,
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 informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationdiffy boxes (iterations of the ducci four number game) 1
diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.
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 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 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 informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationSection 3 Sequences and Limits
Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the nth term of the sequence.
More informationProblem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB
Problem Set 5 1. (a) Fourdigit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11
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 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 informationIntroduction to Diophantine Equations
Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More information1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real
More informationSTRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary
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 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 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 informationSection 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider
More information4. Number Theory (Part 2)
4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.
More information10 k + pm pm. 10 n p q = 2n 5 n p 2 a 5 b q = p
Week 7 Summary Lecture 13 Suppose that p and q are integers with gcd(p, q) = 1 (so that the fraction p/q is in its lowest terms) and 0 < p < q (so that 0 < p/q < 1), and suppose that q is not divisible
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 informationCourse Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.
Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have
More informationSteps to Proving a Theorem
Steps to Proving a Theorem Step 1: Understand Goal What am I looking for? What should the last sentence in my proof say? What are some equivalent ways to state what I want? Are there any complicated definitions
More informationPythagorean Triples Pythagorean triple similar primitive
Pythagorean Triples One of the most farreaching problems to appear in Diophantus Arithmetica was his Problem II8: To divide a given square into two squares. Namely, find integers x, y, z, so that x 2
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 information