# APPLICATIONS OF THE ORDER FUNCTION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and LCMs, a method of showing certain numbers are irrational, and counting the number of zeros at the end of n!. 1. Review We begin with a review of results from the previous lecture. Definition 1. If m is a nonzero integer and if p is a prime, then Ord p m) is the largest integer k such that p k divides m. Proposition 2. Let p be a prime and m be a nonzero integer. Then p m if and only if Ord p m) > 0. Proposition 3. Let m be a nonzero integer, and p be a prime. Then Ord p m) = k if and only if m = up k for some u relatively prime to m. Proposition 4. If p is a prime and u is an integer such that p u, then Ord p up k ) = k. Proposition 5. Let p be a prime. If a, b Z are non-zero then Ord p ab) = Ord p a) + Ord p b). More generally, if each a 1,..., a n Z is nonzero then Ord p a 1 a n ) = Ord p a 1 ) Ord p a n ). Proposition 6. If a Z is non-zero and if k 0 then Ord p a k ) = k Ord p a). Proposition 7. Suppose p 1,..., p k are distinct primes. Then Ord p p m 1 1 p m k k ) = { mi if p = p i 0 if p {p 1,..., p k } Proposition 8 Product formula). Let n be a positive integer, and let p 1,..., p k be a finite sequence of distinct primes that includes every prime divisor of n. Then n = p Ordp i n) i. Exercise 1. Let n be a positive integer. Show that n is a perfect square if and only if 2 Ord p n) for all primes p. Hint: one direction uses the product formula. Date: Spring Version of March 5,

2 2. Additional Results Here are a few useful formulas involving order. Proposition 9. If n Z is nonzero, and p is a prime, then Ord p n) = Ord p n). Proposition 10. Suppose m and n are positive integers. primes p, then m = n. If Ord p m) = Ord p n) for all Proof. Let p 1,..., p k be a finite sequence of distinct primes that include all primes dividing m and n. By the product formula n = p Ordp i n) i = p Ordp i m) i = m. The connection between the order functions and divisibility is described by the following theorem. Theorem 11. Suppose m, n Z are both non-zero. Then m n if and only if Ord p m) Ord p n) for all primes p. Remark. Since Ord p m) = Ord p n) = 0 for all p not dividing m and n, we only need to check the right-hand condition for p dividing m or dividing n. Proof. Suppose m n. Then n = ml for some l Z. So Ord p n) = Ord p m) + Ord p l) for all primes p. This implies that Ord p m) Ord p n) for all primes p. Now suppose that Ord p m) Ord p n) for all primes p. First assume that m and n are positive. Let p 1,..., p k be a finite sequence of distinct primes that includes all prime divisors of m and n. Let a i = Ord pi m) and b i = Ord pi m). By assumption, a i b i. Define l Z by the formula l def = p b i a i i By Proposition 7, Ord pi l) = b i a i. Thus, for all p i in the sequence p 1,..., p k, Ord pi ml) = Ord pi m) + Ord pi l) = a i + b i a i ) = b i = Ord pi n). For p not in the sequence, Ord p ml) = Ord p m) + Ord p l) = = 0 = Ord p n). By Proposition 10, ml = n. So m n as desired. If m and n are not both positive, then Ord p m ) Ord p n ) for all primes p Proposition 9). Thus m n by the above argument. But m n implies that m n. Corollary 12. Suppose a, b, d Z are non-zero. Then d is a common divisor of a and b if and only if Ord p d) min Ord p a), Ord p b)) for all primes p. 2

3 Corollary 13. Suppose a and b are non-zero integers. Let p 1,..., p k be a finite sequence of distinct primes that include all divisors of a and b. Then GCDa, b) = where n i = min Ord pi a), Ord pi b)). Proof. Let g = k pn i i. Proposition 7 and definition of n i, p n i i Ord pi g) = n i = min Ord pi a), Ord pi b)) for all p i in the finite sequence, and Ord p g) = 0 for other p. Thus Ord p g) = min Ord p a), Ord p b)) for all primes p. By Corollary 12, g is a common divisor. Suppose d is any other positive divisor. By Corollary 12, Ord p d) min Ord p a), Ord p b)) = Ord p g) for all primes p. Thus d g by Theorem 11. Thus d g. This shows that g is the greatest common divisor. 3. Least common multiplies LCM) We start with a criterion for a number to be a common multiple. Proposition 14. Suppose a, b, m Z are non-zero. Then m is a common multiple of a and b if and only if Ord p m) max Ord p a), Ord p b)) for all primes p. Proof. If m is a common multiple of a and b, then a m and b m. By Theorem 11, this means that Ord p a) Ord p m) and Ord p b) Ord p m) for all primes p. In other words, Ord p m) max Ord p a), Ord p b)) for all primes p. If Ord p m) max Ord p a), Ord p b)) for all primes p, then Ord p a) Ord p m) and Ord p a) Ord p m). By Theorem 11, m is a common multiple of a and b. Theorem 15. Suppose a and b are non-zero integers. Then there is a unique least common positive multiple LCM) M of a and b. Furthermore, an integer c is a common multiples of a and b if and only if c is a multiple of M. The least common positive) multiple M is given by the formula LCMa, b) = where p 1,..., p k is a finite sequence of distinct primes that include all divisors of a and b. and n i = min Ord pi a), Ord pi b)). Proof. Let M = k pn i i where n i and p i are as above. We will show that M has all the desired properties. Obviously M is positive closure under multiplication). By Proposition 7 and definition of n i, Ord pi M) = n i = max Ord pi a), Ord pi b)) for all p i in the finite sequence, and Ord p g) = 0 for other p. Thus p n i i Ord p M) = max Ord p a), Ord p b)) 3

4 for all primes p. By the previous proposition, M is a common multiple. Suppose c is any other common multiple. By Corollary 12, Ord p c) max Ord p a), Ord p b)) = Ord p M) for all primes p. Thus M c by Theorem 11. A similar argument shows that if M c then c is a common multiple. So c is a common multiple of a and b if and only if c is a multiple of M. In particular, if c is a positive common multiple, then M c, so M c. Thus M is smaller than any other common multiple. So M is a least common multiple. Uniqueness is obvious M M and M M implies M = M ). Here is an interesting formula relating LCMa, b) and GCDa, b) : Theorem 16. Let a and b be positive integers. Then LCMa, b) GCDa, b) = ab Proof. For any prime p, let m p = Ord p a) and n p = Ord p b). Then, by Corollary 13, Theorem 15, and Proposition 7, Ord p GCDa, b) ) + Ordp LCMa, b) ) = minmp, n p ) + maxm p, n p ). By the following lemma, minm p, n p ) + maxm p, n p ) = m p + n p = Ord p a) + Ord p b) Thus LCMa, b) GCDa, b) and ab have the same order for all p). By Proposition 10, they are equal. Lemma 17. Let m, n Z. Then minm, n) + maxm, n) = m + n. Exercise 2. Prove the above lemma. Remark. This lemma does not use any special properties of Z. In fact, it is true of m, n U where U is any linearly ordered set, and + is any commutative binary operation on U. This above theorem generalizes: Proposition 18. Let a and b be non-zero integers. Then LCMa, b) GCDa, b) = ab. Proof. Apply Theorem 16 to a and b. Then use the following lemma. Lemma 19. Let a and b be non-zero integers. Then LCMa, b) = LCM a, b ), GCDa, b) = GCD a, b ). Exercise 3. Prove the above by showing that a and b have the same common divisors and multiples) as a and b. 4

5 4. Irrationality Theorem In this section we will see that n 1/k is irrational unless n is a kth power. For example, 5 is irrational since 5 is not a square, 4 1/3 is irrational since 4 is not a cube, 10 1/5 is irrational since 10 is not a 5th power. We will also see an example of how to show certain logarithms are irrational. One problem with these results is that they are not phrased in terms of integers. They appeal to certain real numbers and assert that these number are not in Q. Fortunately we can convert these results into number theory problems. Begin by supposing that n 1/k is rational, and can be written as a/b. Then n 1/k b = a. Thus nb k = a k. In other words, the equation ny k = x k has a solution with positive x, y. Conversely, any solution to to ny k = x k with x, y positive integers gives a fraction x/y for the root n 1/k. So we can rephrase the problem of showing n 1/k is irrational to the problem of showing that ny k = x k has no solution with x, y positive integers. We begin with a couple illustrations. Then we consider the general theorem. Theorem 20. The equation x 2 = 5y 2 has no positive integer solutions. In other words, 5 is irrational. Proof. Suppose otherwise, that 5y 2 = x 2 has solution x 0, y 0. So x 2 0 = 5y 2 0. Thus This implies that Ord 5 x 2 0) = Ord 5 5y 2 0). 2Ord 5 x 0 ) = Ord 5 5) + 2 Ord 5 y 0 ) = 1 + 2Ord 5 y 0 ) The left hand side is an even integer, the right hand side is an odd integer. Contradiction. Theorem 21. The equation x 3 = 2y 3 has no positive integer solutions. In other words, 2 1/3 is irrational. Proof. Suppose otherwise, that x 3 = 2y 3 has solution x 0, y 0. So x 3 0 = 2y 3 0. Thus This implies that Ord 2 x 3 0) = Ord 2 2y 3 0). 3 Ord 2 x 0 ) = Ord 2 2) + 3 Ord 2 y 0 ) = Ord 2 y 0 ) The left hand side is divisible by three, but the right hand side is not. Contradiction. Theorem 22. Suppose n is a positive integer and that n is not of the form m k with m Z. Then ny k = x k has no positive integer solutions. In other words, n 1/k is irrational unless n is a kth power. Proof. Suppose otherwise, that ny k = x k has solution x 0, y 0. Let p be a prime. Then Ord p ny k 0) = Ord p x k 0). Thus Ord p n) + k Ord p y 0 ) = k Ord p x 0 ). This implies that Ord p n) 0 mod k By the following lemma, this implies that n is a kth power, contradicting the hypothesis. 5

6 Lemma 23. Let n be a positive integer. Then n is a kth power if and only if k Ord p n) for all primes p. Proof. If n = m k then Ord p n) = k Ord k m) Proposition 6). Thus k Ord p n) for all p. Conversely, suppose k Ord p n) for all primes p. Let p 1,..., p q be a finite sequence of distinct primes that include all primes dividing n. Then Ord pi n) = ka i for some nonnegative integer a i since k Ord pi n)). By the product formula, q q n = p ka i i = p a i ) q ) k k i = p a i i. Thus n is a kth power. Now we will see why log 10 2) is irrational. Of course, this generalizes to other logarithms as well. To do so, we convert the problem into a diophantine equation. If log 10 2) = x 0 /y 0 for integers x 0, y 0 with y 0 0, then by definition of logarithms 10 x 0/y 0 = 2. In other words, 10 x 0 = 2 y 0. This implies that 10 x = 2 y has a solution with y 0. Conversely, if such a solution exists, then log 10 2) is rational. We show this cannot happen. Theorem 24. The equation 10 x = 2 y has no solution with x, y Z and y 0. In other words, log 10 2) is irrational. Proof. Suppose that 10 x 0 = 2 y 0 where y 0 0. Apply Ord 5 to both sides: x 0 Ord 5 10) = y 0 Ord 5 2) But Ord 5 10) = 1 and Ord 5 2) = 0. Thus x 0 = 0. This implies that 10 0 = 2 y. So 2 y = 1. Apply Ord 2 to both sides. This implies that yord 2 2) = Ord 2 1). But Ord 2 2) = 1 and Ord 2 1) = 0. Thus y 0 = 0, a contradiction. 5. Zeros of n! How many zeros are at the end of n!? For example, 15! = has three zeros at the end of its decimal expansion, and 50! = has 12 zeros. The number of zeros at the end of a number s decimal expansion is just the largest power of ten dividing that number. Lemma 25. The largest power of 10 dividing n is min Ord 2 n), Ord 5 n) ). Exercise 4. Prove the above. To use the above formula it helps to have a formula for Ord p n!). Lemma 26. If p is a prime, and n is a positive integer, then n Ord p n!) = Ord p k). k=1 6

7 Proof. Use the formula n! = n k, and apply Proposition 5. k=1 Exercise 5. How many zeros does 56! have? How many zeros does 231! have? Hint: Ord 2 n!) Ord 5 n!) so it is enough to compute Ord Exercises Here are some exercises that can be solved with the order functions Ord p. They can also be solved with the unique factorization theorem, but the solutions using the order functions should be a bit easier). Exercise 6. Show that if a k b k then a b. Exercise 7. Show that if 3a 3 b c 2 then a c. Exercise 8. Show that if a 5 c b 4 then a b. Exercise 9. Show that if ab is a square, and if a and b are relatively prime, then a and b are squares. Give a counter example when a and b are not relatively prime. Exercise 10. Show that if p a k where p is a prime, then p a. Exercise 11. Show that 5 2/3 is irrational. 7

### 2 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

### 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.

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,

### Homework 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

### Homework 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

### Chapter 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,

### Discrete 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

### Continued 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

### GREATEST 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

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Prime 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

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### 4. 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.

### Algebra 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

### Elementary 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

### Today 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

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### It 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

### Stanford 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,

### 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,

### CONTINUED 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.

### Practice 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.-

### Continued 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

### Lecture 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 non-negative integers. We say that A divides B, denoted

### Mathematical 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,

### 12 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

### MATH10040 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

### CHAPTER 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

### MATH 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

### Course 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

### 10 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

### INTRODUCTION 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

### k, 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

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### MATH 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

### Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5 1. (a) Four-digit 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

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### Handout 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

### MTH 382 Number Theory Spring 2003 Practice Problems for the Final

MTH 382 Number Theory Spring 2003 Practice Problems for the Final (1) Find the quotient and remainder in the Division Algorithm, (a) with divisor 16 and dividend 95, (b) with divisor 16 and dividend -95,

### Integer 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

### MATHEMATICAL 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

### 8 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

### Further 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

### MAT2400 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

### We 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

### CLASS 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

### Course 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

### SUM 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

### GCDs 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

### Mathematical 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

### Kevin 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

### it 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

### Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2)

Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2) Problems from 21 211 Prove that a b (mod n) if and only if a and b leave the same remainder when divided by n Proof Suppose a

### The 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,

### 1. 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

### The 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,

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### U.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

### On 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

### Lecture 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

### Applications 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

### 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.

### 3 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

### STRAND 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

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

### Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com

Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up 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)).

### 1 Die hard, once and for all

ENGG 2440A: Discrete Mathematics for Engineers Lecture 4 The Chinese University of Hong Kong, Fall 2014 6 and 7 October 2014 Number theory is the branch of mathematics that studies properties of the integers.

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Math 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,

### Section 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

### Practice Problems Solutions

Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

### 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

### Solutions to Assignment 4

Solutions to Assignment 4 Math 412, Winter 2003 3.1.18 Define a new addition and multiplication on Z y a a + 1 and a a + a, where the operations on the right-hand side off the equal signs are ordinary

### SECTION 10-2 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

### Proof: 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

### Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II

Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II Last week: How to find one solution to a linear Diophantine equation This week: How to find all solutions to a linear Diophantine

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### Case #1 Case #2 Case #3. n = 3q n = 3q + 1 n = 3q + 2

Problems 1, 2, and 3 are worth 15 points each (5 points per subproblem). Problems 4 and 5 are worth 30 points each (10 points per subproblem), for a total of 105 points possible. 1. The following are from

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### MOP 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

### Discrete 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

### Continued 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

### Algebraic Systems, Fall 2013, September 1, 2013 Edition. Todd Cochrane

Algebraic Systems, Fall 2013, September 1, 2013 Edition Todd Cochrane Contents Notation 5 Chapter 0. Axioms for the set of Integers Z. 7 Chapter 1. Algebraic Properties of the Integers 9 1.1. Background

### Algebraic and Transcendental Numbers

Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

### This 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

### Unit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013

Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors

### 2.2 Inverses and GCDs

2.2. INVERSES AND GCDS 49 2.2 Inverses and GCDs Solutions to Equations and Inverses mod n In the last section we explored multiplication in Z n.wesaw in the special case with n =12and a =4that if we used

### THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

### Quotient 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.

### PYTHAGOREAN 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 =

### Induction. Mathematical Induction. Induction. Induction. Induction. Induction. 29 Sept 2015

Mathematical If we have a propositional function P(n), and we want to prove that P(n) is true for any natural number n, we do the following: Show that P(0) is true. (basis step) We could also start at

### PROBLEM 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

### 3. 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

### CHAPTER 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,

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### Inference Rules and Proof Methods

Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument

### Math 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