Number Theory. Introduction
|
|
- Lee Atkinson
- 5 years ago
- Views:
Transcription
1 Number Theory Introduction Number theory is the branch of algebra which studies the properties of the integers. While we may from time to time use real or even complex numbers as tools to help us study the integers, these other number systems are not our primary focus. You might think that the integers are simpler than the real or complex numbers, and so number theory should be an easy thing to study. This is not true. Number theory is one of the deepest and most difficult branches of mathematics. We will barely scratch the surface. We will begin by studying simple equations involving integers, and trying to solve them. Consider an equation like 21x + 45y = 12. We want to find all integer values of x and y that make this equation true. This is a very simple looking equation. If we allowed x and y to take real values, it would give the equation of a line, which we could graph. We could find a formula for y in terms of x, and describe the set of all solutions quite precisely. However if we restrict ourselves to integer solutions only, none of this is any help. Algebraic equations involving the integers which must be solved in the integers are called Diophantine equations after a famous dead Greek bloke called Diophantus, who fiddled around with this kind of thing a lot. The equation above is an example of a linear Diophantine equation. We will find a method of solving such equations completely. Things get a lot harder however when the equations stop being linear. Quadratic or cubic equations involving the integers can be most difficult to solve. Frequently there are no integer solutions to such equations, and the task may be to prove this. Divisibility and the Division process In the integers we may add subtract and multiply freely, however we may not always divide. If we try to divide one integer by another, we are only guaranteed of being able to do the division up to a remainder. We know from primary school how to divide integers leaving a remainder. We know that it is always possible, and that there is only one possible answer, no matter how you do it. Let us sum up what we know. The Division Process If a and b are two integers with b 0, then we may find two other integers q (the quotient) and r (the remainder) with 0 r < b so that a = qb + r 1
2 Furthermore q and r are unique with this property. (There is only one possible answer). As you read this, make sure you agree that this is what division with remainder does. Also check out what happens when a and b are negative, and make sure that everything works OK. In the case that r = 0, then a = qb and we say that b divides a and write b a to denote this. There are other equivalent phrases in English which describe this relationship. We can say that b is a factor of a, or that a is a multiple of b. If b a then ±b ± a. For the purposes of divisibility, negative signs are irrelevant. Sometimes because of this we may think only about positive numbers, knowing that the negative case is the same. Note that if b a and a 0 then b a. Hence each non-zero number has only a finite number of factors. For example the factors of 14 are { 14, 7, 2, 1, 1, 2, 7, 14}. Every number a has at least the factors { a, 1, 1, a}. Numbers for which these are the only factors are called prime numbers. Note that negative numbers can be prime, and in fact a is prime if and only if a is prime. Zero is clearly not prime. From this definition, 1 and 1 would appear to be prime, however we choose to specifically exclude them from the definition. This is because 1 and 1 have properties quite unlike the other prime numbers, and if we allowed them to be prime, we would be forever mentioning them as special cases. We call these numbers units instead. Note that units are precisely those numbers that divide every other number. If a and b are two integers, then we call another integer c a common factor of a and b if c a and c b. In fact if {a 1, a 2,..., a n } are integers, we call c a common factor of the set {a 1, a 2,..., a n } if c a i for all i = 1, 2,..., n. Since the units are factors of everything, common factors always exist. Consider a Diophantine equation ax + by = k. If c is a common factor of a and b, then c will be a factor of ax + by for any choice of integers for x and y. So the only way we can get solutions to the Diophantine equation is if c k. For example the Diophantine equation 21x + 35y = 12 has no solutions, since 7 will always be a factor of the left hand side, and yet 7 is not a factor of 12. The problem of solving linear Diophantine equations is thus closely tied up with the problem of finding common factors. The Euclidean Algorithm In this section we present a very useful algorithm for finding common factors of two integers a and b. The algorithm is however much more useful than just that. From it, we will be able to (with a little bit of work) read off solutions to all Diophantine equations of 2
3 the form ax+by = k. Furthermore, the Euclidean algorithm is used in proving many nontrivial properties of the integers. It is our most important algebraic tool when working with the intergers, and has many applications. Let a and b be two given integers with b 0. Divide (with remainder) a by b. We obtain integers q and r with 0 r < b so that a = qb + r Let us look at this equation for a minute. We can even rewrite it as r = a qb. Suppose that c is a common factor of a and b. Then c (a qb) and so c r. Hence c is a common factor of b and r. Conversely, suppose that c is a common factor of b and r. Then c (qb + r) and so c a. Hence c is a common factor of a and b. We see from this, that the common factors of a and b are exactly the same as the common factors of b and r. So instead of looking for common factors of a and b, we can find the common factors of b and r instead. Why bother? you might ask. Well usually r (the remainder) is a lot smaller than a, and hence finding common factors is a lot easier if we use r instead of a. Now the key to the whole thing is the realisation that we need not stop there. If we do it again and again, at each step the numbers will get smaller, but the common factors remain the same. This is essentially all that the Euclidean algorithm involves. a = q 1 b + r 1 0 r 1 < b b = q 2 r 1 + r 2 0 r 2 < r 1 r 1 = q 3 r 2 + r 3 0 r 3 < r 2 r 2 = q 4 r 3 + r 4 0 r 4 < r 3... r n 1 = q n+1 r n + r n+1 0 r n+1 < r n Note that at each step, the quotient is ignored, and the remaining numbers are shifted to the left. Note also that the remainders form a strictly decreasing sequence of positive integers. Thus if we continue the process long enough, at some point we will end up with r n+1 = 0, which is when the process will stop. The last equation will then be r n 1 = q n+1 r n So we see that r n is a factor of r n 1, and hence the common factors of r n and r n 1 will be precisely the factors of r n. We then argue that the factors of r n are the common factors of r n and r n 1, which are the common factors of r n 1 and r n 2, which are the common factors of r n 2 and r n 3, which are... etc, etc... the common factors of a and b. 3
4 Hence the common factors of a and b will consist of all the factors of r n. Notice that as an important consequence of this, the common factors of two integers a and b are always the factors of a single number. Such a number is called a greatest common divisor, or g.c.d. for short. We call d a g.c.d. of a and b if whenever c a and c b then c d. Each pair of numbers has two greatest common divisors, a positive one and a negative one. The Euclidean algorithm always gives us the positive one. We denote this positive greatest common divisor of a and b by the expression (a, b). It is a bit like the situation with regard to square roots, where x always denotes the positive square root even though two square roots, a positive one and a negative one, exist. It is time for an example. Consider the two integers and We apply the Euclidean algorithm below. Note that our two initial numbers and our remainders are underlined. This is to remind us of which numbers we are working with during the calculation. We will see later another reason for this. We thus see that (10362, 12397) = = = = = = = 2 11 Now consider for a moment the Diophantine equation 10362x y = k for some fixed constant k. As 11 is always a factor on the left hand side, there will be no solution to the equation unless k is a multiple of 11. Let us consider for a moment the case that k = 11. This is the smallest interesting equation where we might hope to find solutions. It turns out that if we can solve this equation, we can solve all the others. I claim that the Euclidean algorithm not only finds the g.c.d. for us, but gives us a solution to this Diophantine equation as well. To obtain this solution we apply a process of back-substitution to the steps in the Euclidean algorithm. We rearrange the steps in the Euclidean algorithm, (all except for the last one) to make the remainder in each case the subject of the equation. In this case we obtain 2035 = = = = =
5 Now we begin with the last equation, and use the other equations working from bottom to top, to substitute for the appropriate underlined expressions. I have performed this calculation below. Note that the underlined numbers are to be regarded as protected, and should not be calculated out at any time. We can and will collect these numbers up as common factors however. And we are free to do any calulations we like with the non-underlined numbers to simplify the resulting expressions. 11 = = ( ) = = 8 ( ) = = ( ) = = 443 ( ) = Hence a solution to the Diophantine equation 10362x y = 11 occurs when x = 530 and y = 443. I hope you will agree that finding such a solution by trial and error without using the Euclidean algorithm would be rather difficult. Let us now sum up what the Euclidean algorithm does for us in a nice concise form. The Euclidean Algorithm Given any two non-zero integers a and b there exist two integers m and n with am + bn = (a, b), and these can be obtained by the Euclidean Algorithm using back-substitution. Linear Diophantine Equations We now have everything we need to solve linear Diophantine equations. Consider the equation ax + by = k We firstly find (a, b) using the Euclidean Algorithm. If k is NOT a multiple of (a, b) then the equation has no integer solutions, and we can just state this fact and stop. If k is a multiple of (a, b) then we can write k = t(a, b) for some integer t. The Euclidean algorithm gives us integers m and n so that am+bn = (a, b). If we multiply both sides of this equation by t we obtain the equation a(tm) + b(tn) = t(a, b) = k 5
6 So we see that x = tm and y = tn gives a solution to the Diophantine equation. For example, the equation 10362x y = 13 has no solutions as 13 is not a multiple of 11. On the other hand, the equation 10362x y = 55 has solutions since 55 = We can obtain a solution by simply multiplying the values obtained from the Euclidean algorithm by 5. This gives us a solution when x = = 2650 and y = = One more example. Consider the equation 10362x y = 0 Now clearly this will have solutions, since everything divides 0. In fact you need not even do the Euclidean algorithm in this case as a solution to this equation is obvious, namely x = 0 and y = 0. The case where the constant term is zero is as you can see, very easy. So far we have discovered how to tell if a linear Diophantine equation has solutions, and in the case when solutions exist, we can always find one of them. What we now would like to do is find if there are any more solutions, and if so give a formula to describe all of them. We will do this in a theoretical way first, and then do an example. Suppose we have the Diophantine equation ax + by = k = t(a, b) And that we have discovered (using the Euclidean Algorithm) that this has a solution when x = x 0 and y = y 0. Thus x 0 and y 0 obey the above equation. We write these two equations down, and subtract. We will then obtain We thus have the equation ax + by = k ax 0 + by 0 = k a(x x 0 ) + b(y y 0 ) = 0 a(x x 0 ) = b(y y 0 ) Now we notice that the left hand side is a multiple of a, and the right hand side is a multiple of b. Therefore, since the two sides are equal, the whole thing is divisible by both 6
7 a and b. Before we can continue, we need to do some work on common multiples. So we will take a break from Diophantine equations for a minute, while we do this. If a and b are two integers, we say that c is a common multiple of a and b if a c and b c. Note that ab is a common multiple of a and b, but there are clearly often smaller common multiples. For example 10 and 45 have common multiple 90, which is smaller than = 450. If c and c are two common multiples of a and b then mc + nc will be a common multiple of a and b for every choice of integers m and n. In particular, we can choose m and n via the Euclidean algorithm, and conclude that (c, c ) is a common multiple of a and b. This tells us that we can find a smaller common multiple which divides the given ones. By repeating this process we see that there must exist a least common multiple or l.c.m. of a and b, and that all other common multiples are multiples of the least common multiple. More formally, the LCM is a number c which is positive and which divides every common multiple of a and b. The argument above can be tidied up into a proof that least common multiples always exist. We denote the least common multiple of a and b by the expression [a, b]. Note that ab is a common multiple, hence it must be a multiple of [a, b]. Suppose [a, b]d = ab. Then [a, b] b and since the left hand side is an integer, so is the right hand side and therefore d divides a. Similarly d divides b and so d is a common divisor. So in particular d (a, b). On the other hand ab /(a, b) is a common multiple of a and b, and so is bigger than [a, b]. That is ab /(a, b) [a, b] which implies (since everything is positive) that (a, b) ab /[a, b] = d. We can conclude that = a d ab = (a, b)[a, b] This simple formula allows us to find the l.c.m. from the g.c.d. and vice versa. Now let us return to the question of solving Diophantine equations. We earlier were considering the equation a(x x 0 ) = b(y y 0 ) This quantity is a common multiple of a and b, and hence is a multiple of [a, b]. Thus x x 0 = t[a, b]/a and y y 0 = t[a, b]/b. 7
8 [a, b] Using the relation [a, b](a, b) = ab we can write = ± b [a, b] and = ± a a (a, b) b (a, b) where the plus or minus sign is the same in each case and depends on whether or not a and b have the same sign. So once we know the gcd, we can work out these numbers without actually having to work out the lcm (which could be quite large) first. Putting all this together we can write the formula for the general solution to the Diophantine equation as ( ) b x = x 0 + t (a, b) ( ) a y = y 0 t (a, b) Where t is an integer. This formula works in all cases of a and b being positive or negative, although in practical terms it is best to write the numbers down and apply a common sense reality check to see that the signs are right for the extra terms to be cancelled in the Diophantine equation. We can obtain any solution we like from this general formula by plugging in different integer values for t. By way of an example, consider the equation 10362x y = 55 which we had earlier. We then found a particular solution occurred when x = 2650 and y = And also that the gcd was 11. Putting these values into the formula we have found, we find ( x = ( y = ) t ) t = t = t Every solution to the Diophantine equation is described by this formula. To get a particular solution, we merely choose some number for t. In some cases we might be interested later in using the formula to look for certain solutions. For example we might want to find the solution giving the smallest positive value for x. Looking above we see that this will happen when t = 3, and this gives the particular solution x = 731 and y = 611. This completes our investigation of the solution of linear Diophantine equations in two variables using the Euclidean Algorithm. 8
Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I
Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I Diophantine equations are equations intended to be solved in the integers. We re going to focus on Linear Diophantine Equations.
More informationDirect Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24
Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using
More informationIntermediate Math Circles February 26, 2014 Diophantine Equations I
Intermediate Math Circles February 26, 2014 Diophantine Equations I 1. An introduction to Diophantine equations A Diophantine equation is a polynomial equation that is intended to be solved over the integers.
More informationChapter 1 A Survey of Divisibility 14
Chapter 1 A Survey of Divisibility 14 SECTION C Euclidean Algorithm By the end of this section you will be able to use properties of the greatest common divisor (gcd) obtain the gcd using the Euclidean
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationCool Results on Primes
Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why
More information22. The Quadratic Sieve and Elliptic Curves. 22.a The Quadratic Sieve
22. The Quadratic Sieve and Elliptic Curves 22.a The Quadratic Sieve Sieve methods for finding primes or for finding factors of numbers are methods by which you take a set P of prime numbers one by one,
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationLecture 2. The Euclidean Algorithm and Numbers in Other Bases
Lecture 2. The Euclidean Algorithm and Numbers in Other Bases At the end of Lecture 1, we gave formulas for the greatest common divisor GCD (a, b), and the least common multiple LCM (a, b) of two integers
More information= 5 2 and = 13 2 and = (1) = 10 2 and = 15 2 and = 25 2
BEGINNING ALGEBRAIC NUMBER THEORY Fermat s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus (third century
More information11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic
11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic Bezout s Lemma Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we
More informationMath 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6
Math 131 notes Jason Riedy 6 October, 2008 Contents 1 Modular arithmetic 2 2 Divisibility rules 3 3 Greatest common divisor 4 4 Least common multiple 4 5 Euclidean GCD algorithm 5 6 Linear Diophantine
More informationAn Introduction to Mathematical Thinking: Algebra and Number Systems. William J. Gilbert and Scott A. Vanstone, Prentice Hall, 2005
Chapter 2 Solutions An Introduction to Mathematical Thinking: Algebra and Number Systems William J. Gilbert and Scott A. Vanstone, Prentice Hall, 2005 Solutions prepared by William J. Gilbert and Alejandro
More informationChapter 5. Number Theory. 5.1 Base b representations
Chapter 5 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 informationHonours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period:
Honours Advanced Algebra Name: Unit : Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period: Mathematical Goals Know and apply the Remainder Theorem Know and apply
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 3
CS 70 Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 3 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a smaller
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 informationFinite Mathematics : A Business Approach
Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know
More informationSection 4. Quantitative Aptitude
Section 4 Quantitative Aptitude You will get 35 questions from Quantitative Aptitude in the SBI Clerical 2016 Prelims examination and 50 questions in the Mains examination. One new feature of the 2016
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More information6: Polynomials and Polynomial Functions
6: Polynomials and Polynomial Functions 6-1: Polynomial Functions Okay you know what a variable is A term is a product of constants and powers of variables (for example: x ; 5xy ) For now, let's restrict
More informationIES Parque Lineal - 2º ESO
UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....
More informationMath Lecture 18 Notes
Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 5
CS 70 Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 5 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a
More informationFactors, Zeros, and Roots
Factors, Zeros, and Roots Mathematical Goals Know and apply the Remainder Theorem Know and apply the Rational Root Theorem Know and apply the Factor Theorem Know and apply the Fundamental Theorem of Algebra
More informationLecture 4: Number theory
Lecture 4: Number theory Rajat Mittal IIT Kanpur In the next few classes we will talk about the basics of number theory. Number theory studies the properties of natural numbers and is considered one of
More informationMath Circles - Lesson 2 Linear Diophantine Equations cont.
Math Circles - Lesson 2 Linear Diophantine Equations cont. Zack Cramer - zcramer@uwaterloo.ca March 7, 2018 Last week we discussed linear Diophantine equations (LDEs), which are equations of the form ax
More informationDivisibility of Natural Numbers
10-19-2009 Divisibility of Natural Numbers We now return to our discussion of the natural numbers. We have built up much of the mathematical foundation for the natural numbers (N = 1, 2, 3,...). We used
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More information1 Continued Fractions
Continued Fractions To start off the course, we consider a generalization of the Euclidean Algorithm which has ancient historical roots and yet still has relevance and applications today.. Continued Fraction
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
More information1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More information32 Divisibility Theory in Integral Domains
3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationDiscrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6
CS 70 Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes
More informationLesson 5b Solving Quadratic Equations
Lesson 5b Solving Quadratic Equations In this lesson, we will continue our work with Quadratics in this lesson and will learn several methods for solving quadratic equations. The first section will introduce
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationQuestionnaire for CSET Mathematics subset 1
Questionnaire for CSET Mathematics subset 1 Below is a preliminary questionnaire aimed at finding out your current readiness for the CSET Math subset 1 exam. This will serve as a baseline indicator for
More informationFermat s Last Theorem for Regular Primes
Fermat s Last Theorem for Regular Primes S. M.-C. 22 September 2015 Abstract Fermat famously claimed in the margin of a book that a certain family of Diophantine equations have no solutions in integers.
More informationChapter 2. Divisibility. 2.1 Common Divisors
Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition
More informationWhat is proof? Lesson 1
What is proof? Lesson The topic for this Math Explorer Club is mathematical proof. In this post we will go over what was covered in the first session. The word proof is a normal English word that you might
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationCongruences. September 16, 2006
Congruences September 16, 2006 1 Congruences If m is a given positive integer, then we can de ne an equivalence relation on Z (the set of all integers) by requiring that an integer a is related to an integer
More informationTAYLOR POLYNOMIALS DARYL DEFORD
TAYLOR POLYNOMIALS DARYL DEFORD 1. Introduction We have seen in class that Taylor polynomials provide us with a valuable tool for approximating many different types of functions. However, in order to really
More informationAn Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Science, Chennai. Unit - I Polynomials Lecture 1B Long Division
An Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Science, Chennai Unit - I Polynomials Lecture 1B Long Division (Refer Slide Time: 00:19) We have looked at three things
More informationAlgebra Year 10. Language
Algebra Year 10 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.
More information#26: Number Theory, Part I: Divisibility
#26: Number Theory, Part I: Divisibility and Primality April 25, 2009 This week, we will spend some time studying the basics of number theory, which is essentially the study of the natural numbers (0,
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationThe Integers. Peter J. Kahn
Math 3040: Spring 2009 The Integers Peter J. Kahn Contents 1. The Basic Construction 1 2. Adding integers 6 3. Ordering integers 16 4. Multiplying integers 18 Before we begin the mathematics of this section,
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
More informationHonors Advanced Algebra Unit 3: Polynomial Functions October 28, 2016 Task 10: Factors, Zeros, and Roots: Oh My!
Honors Advanced Algebra Name Unit 3: Polynomial Functions October 8, 016 Task 10: Factors, Zeros, and Roots: Oh My! MGSE9 1.A.APR. Know and apply the Remainder Theorem: For a polynomial p(x) and a number
More informationNumber Sense. Basic Ideas, Shortcuts and Problems #1-20 from the Sequence Chart
UIL Number Sense Contest Basic Ideas, Shortcuts and Problems #1-20 from the Sequence Chart Larry White UIL State Number Sense Contest Director texasmath@centex.net http://www.uiltexas.org/academics/number-sense
More informationContribution of Problems
Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
More informationThe Euclidean Algorithm and Diophantine Equations
The Euclidean Algorithm and Diophantine Equations Donald Rideout, Memorial University of Newfoundland 1 Let Z = {0, ±1, ±2, } denote the set of integers. For a, b Z, a 0, we say b is divisible by an integer
More informationTopic Contents. Factoring Methods. Unit 3: Factoring Methods. Finding the square root of a number
Topic Contents Factoring Methods Unit 3 The smallest divisor of an integer The GCD of two numbers Generating prime numbers Computing prime factors of an integer Generating pseudo random numbers Raising
More informationQUADRATIC EQUATIONS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Quadratic Equations Page 1 of 8 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier QUADRATIC EQUATIONS Version: 3.1 Date: 6-10-014 Mathematics Revision Guides
More informationIntermediate Algebra Textbook for Skyline College
Intermediate Algebra Textbook for Skyline College Andrew Gloag Anne Gloag Mara Landers Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable
More informationPartial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254
Partial Fraction Decomposition Honors Precalculus Mr. Velazquez Rm. 254 Adding and Subtracting Rational Expressions Recall that we can use multiplication and common denominators to write a sum or difference
More informationChapter Usual types of questions Tips What can go ugly. and, common denominator will be
C3 Cheat Sheet Chapter Usual types of questions Tips What can go ugly 1 Algebraic Almost always adding or subtracting Factorise everything in each fraction first. e.g. If denominators Blindly multiplying
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationElementary Properties of the Integers
Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of
More information6.4 Division of Polynomials. (Long Division and Synthetic Division)
6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator
More informationthen the hard copy will not be correct whenever your instructor modifies the assignments.
Assignments for Math 2030 then the hard copy will not be correct whenever your instructor modifies the assignments. exams, but working through the problems is a good way to prepare for the exams. It is
More informationFactors, Zeros, and Roots
Factors, Zeros, and Roots Solving polynomials that have a degree greater than those solved in previous courses is going to require the use of skills that were developed when we previously solved quadratics.
More informationU7 - Polynomials. Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2.
U7 - Polynomials Name 1 Polynomial Functions - In this first section, we will focus on polynomial functions that have a higher degree than 2. - A one-variable is an expression that involves, at most, the
More informationLinear Diophantine Equations
Parabola Volume 49, Issue 2 (2013) Linear Diophantine Equations David Angell 1 Many popular and well known puzzles can be approached by setting up and solving equations. Frequently, however, it will be
More informationNumber Theory in Problem Solving. Konrad Pilch
Number Theory in Problem Solving Konrad Pilch April 7, 2016 1 Divisibility Number Theory concerns itself mostly with the study of the natural numbers (N) and the integers (Z). As a consequence, it deals
More information3 The fundamentals: Algorithms, the integers, and matrices
3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers
More information4 Number Theory and Cryptography
4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.
More informationFactoring Algorithms Pollard s p 1 Method. This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors.
Factoring Algorithms Pollard s p 1 Method This method discovers a prime factor p of an integer n whenever p 1 has only small prime factors. Input: n (to factor) and a limit B Output: a proper factor of
More informationPolynomials. This booklet belongs to: Period
HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationSECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
More informationTESTCRACKER CAT TOPPER s PROGRAM: QUANT. Factorial
TESTCRACKER CAT TOPPER s PROGRAM: QUANT Factorial The product of n consecutive natural numbers starting from 1 to n is called as the factorial n. n! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x x (n 2) x (n 1) x n e.g.
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationThe Berlekamp algorithm
The Berlekamp algorithm John Kerl University of Arizona Department of Mathematics 29 Integration Workshop August 6, 29 Abstract Integer factorization is a Hard Problem. Some cryptosystems, such as RSA,
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationAlgebra Introduction to Polynomials
Introduction to Polynomials What is a Polynomial? A polynomial is an expression that can be written as a term or a sum of terms, each of which is the product of a scalar (the coefficient) and a series
More informationLearning Packet. Lesson 5b Solving Quadratic Equations THIS BOX FOR INSTRUCTOR GRADING USE ONLY
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationAlgebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review
Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,
More informationMath 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS
4-1Divisibility Divisibility Divisibility Rules Divisibility An integer is if it has a remainder of 0 when divided by 2; it is otherwise. We say that 3 divides 18, written, because the remainder is 0 when
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 8 February 1, 2012 CPSC 467b, Lecture 8 1/42 Number Theory Needed for RSA Z n : The integers mod n Modular arithmetic GCD Relatively
More informationPlease bring the task to your first physics lesson and hand it to the teacher.
Pre-enrolment task for 2014 entry Physics Why do I need to complete a pre-enrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationChapter 3 Basic Number Theory
Chapter 3 Basic Number Theory What is Number Theory? Well... What is Number Theory? Well... Number Theory The study of the natural numbers (Z + ), especially the relationship between different sorts of
More informationFall 2017 Test II review problems
Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and
More informationExample: This theorem is the easiest way to test an ideal (or an element) is prime. Z[x] (x)
Math 4010/5530 Factorization Theory January 2016 Let R be an integral domain. Recall that s, t R are called associates if they differ by a unit (i.e. there is some c R such that s = ct). Let R be a commutative
More informationIn this unit we will study exponents, mathematical operations on polynomials, and factoring.
GRADE 0 MATH CLASS NOTES UNIT E ALGEBRA In this unit we will study eponents, mathematical operations on polynomials, and factoring. Much of this will be an etension of your studies from Math 0F. This unit
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More information1.1.1 Algebraic Operations
1.1.1 Algebraic Operations We need to learn how our basic algebraic operations interact. When confronted with many operations, we follow the order of operations: Parentheses Exponentials Multiplication
More information2. FUNCTIONS AND ALGEBRA
2. FUNCTIONS AND ALGEBRA You might think of this chapter as an icebreaker. Functions are the primary participants in the game of calculus, so before we play the game we ought to get to know a few functions.
More informationRational Numbers CHAPTER. 1.1 Introduction
RATIONAL NUMBERS Rational Numbers CHAPTER. Introduction In Mathematics, we frequently come across simple equations to be solved. For example, the equation x + = () is solved when x =, because this value
More information