# PYTHAGOREAN TRIPLES M. SUNIL R. KOSWATTA HARPER COLLEGE, ILLINOIS

Size: px
Start display at page:

Download "PYTHAGOREAN TRIPLES M. SUNIL R. KOSWATTA HARPER COLLEGE, ILLINOIS"

## Transcription

1 PYTHAGOREAN TRIPLES M. SUNIL R. KOSWATTA HARPER COLLEGE, ILLINOIS Astract. This talk is ased on a three week summer workshop (Los Angeles 2004) conducted y Professor Hung-Hsi Wu, University of California, Berkeley, on Algera for K-12 teachers. Professor Wu has een conducting workshops for K-12 teacher in many parts of the country for several years. He is a memer of the National Math Panal. Three positive integers a,, c form a Pythagorean triple {a,, c} if a = c 2. In other words, a, and c are lengths of three sides of a right triangle. Almost everyody know that {3, 4, 5}, {5, 12, 13}, and {8, 15, 17} are Pythagorean triples. But are there others? Pythagorean triples can e produced at will y solving an extremely simple linear system of equations. It can e shown that this method produces all the Pythagorean triples. One would like to say that this method is due to Baylonians some 3800 years ago, ut a more accurate statement would e that it is the algeraic rendition of the method one infers from a close reading of the cuneiform talet, Plimpton which listed fifteen Pythagorean triples. Lest you entertain for even a split second the idle thought that people couldn t have known such advanced mathematics thirty-eight centuries ago and that these triples were proaly hit upon y trial and error, let it e noted that the largest triple (in Plimpton 322) is {12709, 13500, 18541}. 2 More amazingly, these were the work of a pupil proaly doing his assigned homework from school. 3 Theorem 1. Let (u, v) e the solution of the linear system u + v = t s u v = s t where s and t are positive integers with s < t. If we write u and v as two fractions with the same denominator, u = c and v = a, then {a,, c} is a Pythagorean triple. 1 From the Plimpton collection of cuneiform talets in the Colomia University 2 Wu 3 Roson 1

2 Pythagorean Triples Page 2 Proof of this theorem is very simple and should e accessile to any one with high school algera ackground. Proof. Multiply corresponding sides of the two equations in the theorem to get (u + v)(u v) = t s s t or u2 v 2 = 1. So with, u = c and v = a, we have ( c ) 2 ( a ) 2 = 1. Multiplying through oth sides of this equality y 2 gives c 2 a 2 = 2 and therefore, a = c 2. Let us use theorem 1 to produce some new Pythagorean triples. EXAMPLE 1: Consider u + v = 2 u v = 1 2 You get 2u = 5 2 so that u = 5 4 y adding the two equations. Eliminating u (y sutracting) from the same two equations you get 2v = 3 2 so that v = 3. Thus, the 4 triple retrieved from this system of linear equations is {3, 4, 5}. EXAMPLE 2: Consider 3 2 u v = 2 3 Adding the two equations gives u = 13. Sustituting u in the second equation gives 12 v = 5. Therefore, y theorem 1, {5, 12, 13} is a Pythagorean triple. 12 EXAMPLE 3: Consider 4 3 u v = 3 4

3 Pythagorean Triples Page 3 Adding the two equations gives u = 25. Sustituting u in the second equation gives 24 v = 7. Therefore, y theorem 1, {7, 24, 25} is a Pythagorean triple. 24 EXAMPLE 4: Consider 69 2 u v = 2 69 Adding the two equations gives u = Sustituting u in the second equation gives 276 v = theorem 1 guarantees that {276, 4757, 4765} is a Pythagorean triple. 276 EXAMPLE 5: Consider u v = Adding the two equations gives u = Sustituting u in the second equation gives v = theorem 1 guarantees that {27000, 25418, 37082} is a Pythagorean triple We define a Pythagorean triple {a,, c} to e primitive if the integers a, and c have no common divisor other than 1. We say that a Pythagorean triple is a multiple of anther Pythagorean triple {a,, c } if there is a positive integer n so that a = na, = n and c = nc. In this terminology, a given Pythagorean triple is either primitive, or is a multiple of a primitive Pythagorean triple. EXAMPLE 6: Consider u + v = 5 Adding the two equations you get 2u = 26 5 u v = 1 5 sutracting) from the same two equations you get 2v = so that u =. Eliminating u (y so that v =. Thus, y 10

4 Pythagorean Triples Page 4 Theorem 3, {10, 25, 26} is a Pythagorean triple. This is not primitive ecause it is a multiple of the primitive Pythagorean triple {5, 12, 13}. However, if we have taken the troule to reduce u = 26 to its lowest terms, then we 10 would otain u = 13 and the primitive triple {5, 12, 13} would e the result. Thus we 5 see that different values of s and t do not always lead to distinct primitive Pythagorean triples. We now explain the genesis of Theorem 1. We assume that we already have a Pythagorean triple {a,, c} and proceed to find what it must e. By assumption, a = c 2. By dividing this equation through y 2, we get (1) ( a ) = ( c ) 2. Let u = c and v = a. Thus u and v are oth fractions and u > v. Now (1) ecomes v = u 2 and therefore, (2) u 2 v 2 = 1. Since u 2 v 2 = (u + v)(u v), we get (u + v)(u v) = 1. Since oth (u + v) and (u v) are known fractions, let s, t e two positive integers with s < t so that u + v = t s. Because (u + v)(u v) = 1, necessarily, u v = s. Therefore, we have: t (3) u + v = t s u v = s t where s and t are positive integers with s < t. From this point of view, Theorem 1 is inevitale.

5 Pythagorean Triples Page 5 We can refine Theorem 1 y directly solving system (3). Adding the two equations, we get 2u = t s + s t = t2 + s 2, and therefore, st u = t2 + s 2. By sustituting u in the second equation of (3), we get, A simple computation gives, v = t2 s 2. ( ) t 2 s 2 2 ( ) t 2 + s =. Multiplying oth sides y () 2, we get (t 2 s 2 ) 2 + () 2 = (t 2 + s 2 ) 2. This shows that if s, t are positive integers and t > s, then {, t 2 s 2, t 2 + s 2 } is a Pythagorean triple. We say that two integers x and y are relatively prime if x and y have no common divisors other than ±1. We will use the notation (x, y) = 1 to indicate that x and y are relatively prime. We use the notation (x, y) = z to indicate that z is the greatest common divisor of x and y. If {a,, c} is a Pythagorean triple and if (a, ) = n then n divides c. Therefore, if a = a n, = n and c = c n then {a,, c } is a triple with (a, ) = 1. In other words, {a,, c } is a primitive triple and {a,, c} is a multiple of {a,, c }. Therefore, we will consider {a,, c} to e a primitive triple for the rest of the discussion. Both a and cannot e even ecause if so, then (a, ) 1. We claim that oth a and cannot e odd either. We can estalish this claim y using proof y contradiction

6 Pythagorean Triples Page 6 method. Assume that oth a and are odd. That is, a = n + 1 and = m + 1 for two even integers n and m. Then That is, a = (n + 1) 2 + (m + 1) 2 = n 2 + m 2 + 2n + 2m + 2 = c 2 c 2 2 = n 2 + m 2 + 2n + 2m Since n and m are even, 4 divides (c 2 2). Now this is the contradiction we sought. c is a positive integer. Therefore, it is either even or odd. If c is even the c 2 is divisile y 4. Therefore, c 2 2 is not divisile y 4. If c is odd, then c 2 1 is divisile y 4. (Assume c = k + 1 for some even integer k. Then c 2 = k 2 + 2k + 1 and c 2 1 = k 2 + 2k.) Therefore, c 2 2 is not divisile y 4. Therefore, if {a,, c} is primitive then a and are of opposite parity. Without loss of generality, lets assume that a is even. Theorem 2. If a,, and c are positive integers, then the most general solution of the equation a = c 2, satisfying the conditions, (a, ) = 1, and a is even is a =, = t 2 s 2, c = t 2 + s 2 where s, t are integers of opposite parity and (s, t) = 1 and t > s > 0. Moreover there is a 1-1 correspondence etween different values of s, t and different values of a,, c. 4 Proof. First assume that {a,, c} is a primitive Pythagorean triple and a is even and is odd. Since is odd, let = m + 1 for some even integer m. Then c 2 = a 2 + m 2 + 2m + 1 and therefore, c 2 is odd and hence, c is odd. Also, (, c) = 1. If not, then (, c) = n for some positive integer n 1. That means, c = nc and = n and c 2 2 is divisile y n 2 and therefore, a is divisile y n. This contradicts the fact that (a, ) = 1. ( Since oth and c are odd, 1(c ) and 1 (c + ) are positive integers. Also, (c ), 1(c + )) = 1. If not, then ( 1 (c ), 1(c + )) = d, for some positive integer d 1. That is, c+ = dx and c = dy for some positive integers x and y and x > y. 2 2 That means, c = d(x + y) and = (x y). This contradicts the fact that (, c) = 1. ( a ) ( ) ( ) 2 c c + Now = The two factors on the right, eing relatively prime, must oth e squares. Let c = s 2 and c + = t 2. Clearly, t > s > 0. Also, (s, t) = 1. If not, then (s, t) = u 2 2 for some postive integer u 1. That is s = us and t = ut for some positive integers s and t. Then (s 2, t 2 ) 1 and this is a contradiction. 4 Hardy and Wright

7 Pythagorean Triples Page 7 Since (t, s) = 1, oth t and s cannot e even. If oth t and s are odd, say t = n + 1 and s = m + 1 for positive even integers n and m and n > m, then = t 2 s 2 = (n m)(n + m + 2) eing product of two even integers is even and that is a contradiction. Therefore, s and t are of opposite parity. Now assume that s and t are two positive integers of opposite parity, (s, t) = 1 and t > s > 0 so that a =, = t 2 s 2, c = t 2 + s 2. Then a = 4s 2 t 2 + (t 2 s 2 ) 2 = (t 2 + s 2 ) 2 = c 2. Hence, {a,, c} is a Pythagorean triple, a is even and is odd. If (a, ) = d for some positive integer d 1, then d divides a, and hence c and therefore, d divides t 2 s 2 and t 2 + s 2. That is, t 2 s 2 = dm and t 2 + s 2 = dn for some positive integers m and n and m > n. That means, 2t 2 = d(n + m) and 2s 2 = d(n m). Therefore, d divides oth 2t 2 and 2s 2. But, clearly, (t 2, s 2 ) = 1 since (t, s) = 1. Therefore, d must e either 1 or 2. But d divides and is odd. Therefore, d is 1 and (a, ) = 1. Finally, if and c are given then s 2, and t 2 are uniquely determined and as a consequence s and t are uniquely determined. References [1] Hung-Hsi Wu, Introduction to School Algera [DRAFT], wu/, June [2] G.H. Hardy and E.M. Wright An Introduction to the Theory of Numers, Oxford University Press. [3] O. Neugeauer The Exact Sciences in Antiquity, Brown University Press. [4] R. Creighton Buck Sherlock Holmes in Baylon, The American Mathematical Monthly, Vol. 87, No. 5. (May, 1980), pp [5] Eleanor Roson Neither Sherlock Holmes nor Baylon : A reassessment of Plimpton 322, Historia Mathematica, 28 (2001), [6] Eleanor Roson Words and Pictures: New Light on Plimpton 322, The American Mathematical Monthly, Feruary (2002),

### Congruent Number Problem

University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases

### QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write

### PYTHAGOREAN 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

### Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.

[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write

### 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

### DigitalCommons@University of Nebraska - Lincoln

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University

### Settling 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 E-Mail address:

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### 4.2 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

### RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES

RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean

### 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

### 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.

### 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

### Pythagorean Triples. becomes

Solution Commentary: Solution of Main Problems: Pythagorean Triples 1. If m = (n + 1), n = [m -1]/ and n+1 = [m +1]/. Then, by substitution, the equation n + (n + 1) = (n+1) m + 1 becomes + m =. Now, because

### 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

### Every 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

### 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

### = 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

### 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

### 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

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### A Direct Method To Generate Pythagorean Triples And Its Generalization To Pythagorean Quadruples And n-tuples

A Direct Method To Generate Pythagorean Triples And Its Generalization To Pythagorean Quadruples And n-tuples Tanay Roy \$ Farjana Jaishmin Sonia Department of Physics Jadavpur University Kolkata 700032,

### Chapter 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

### 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

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### 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.

### Stupid 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

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### 15 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

### HOMEWORK 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

### 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

### Introduction to fractions A fraction is the ratio of two whole numbers, i.e. one whole number divided by another whole number:

Fractions & Percentages Topics Covered: Fractions Simplifying fractions Equivalent fractions Improper fractions & mixed numers Operations with Fractions (addition, sutraction, multiplication, division)

### Pythagorean Theorem: Proof and Applications

Pythagorean Theorem: Proof and Applications Kamel Al-Khaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN E-mail: kamel@just.edu.jo,

### TEXAS 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

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### 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

### 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

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### 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

### 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

### Math 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

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### I. 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

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### arxiv:0909.4913v2 [math.ho] 4 Nov 2009

IRRATIONALITY FROM THE BOOK STEVEN J. MILLER AND DAVID MONTAGUE arxiv:0909.4913v2 [math.ho] 4 Nov 2009 A right of passage to theoretical mathematics is often a proof of the irrationality of 2, or at least

### Lesson 9: Radicals and Conjugates

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### ALGEBRAIC 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 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### PRIME FACTORS OF CONSECUTIVE INTEGERS

PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

### Counting Primes whose Sum of Digits is Prime

2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey

### 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

### 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

### 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

### Polynomials - Exponent Properties

5.1 Polynomials - Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with expoenents can often e simplified using a few asic exponent properties. Exponents represent

### Grade 7/8 Math Circles Fall 2012 Factors and Primes

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole

### Handout #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

### 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

### 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

### 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

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

### Cubes and Cube Roots

CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Grade 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

### Lesson 9: Radicals and Conjugates

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

### 3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼

cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The

### LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton

LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton 1. Take a two-digit number and write it down three times to form a six-digit number. For example,

### Linear Equations in One Variable

Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms

The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about

### 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

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### 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

### Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

### ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS

ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number

### Copy 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,...}

### 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,

### Number 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

### Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

### DIVISIBILITY 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

### An Introduction to Number Theory Prime Numbers and Their Applications.

East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

### FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant

### 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

### CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association

CISC - Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### WHEN DOES A CROSS PRODUCT ON R n EXIST?

WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what

### 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,

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### 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

### 10/14/08 ON NUMBERS EQUAL TO THE SUM OF TWO SQUARES IN MORE THAN ONE WAY

1 10/14/08 ON NUMBERS EQUAL TO THE SUM OF TWO SQUARES IN MORE THAN ONE WAY Ming-Sun Li, Kathryn Robertson, Thomas J. Osler, Abdul Hassen, Christopher S. Simons and Marcus Wright Department of Mathematics

### 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

### 10.1 Systems of Linear Equations: Substitution and Elimination

726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

### Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

### 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,