Lecture 2: Continued fractions, rational approximations
|
|
- John Hood
- 5 years ago
- Views:
Transcription
1 Lecture 2: Continued fractions, rational approximations Algorithmic Number Theory (Fall 204) Rutgers University Swastik Kopparty Scribe: Cole Franks Continued Fractions We begin by calculating the continued fraction representation of a real number Let α be a real number Then α = α + α, α = α + α 2, α n = α n + α n where α i is greater than one for i {,, n} Let a 0 := α, and a i := α i At some point α N may be an integer, in which case this process terminates and α is rational In that case, we simply define a n = 0 for all n > N, and α n is undefined for n > N Note that a n for all n N If α is irrational, the process will never terminate, so we don t have to worry about the two previous lines Note that a i, α i 0 for all i By plugging each equation into the one above, we see that α = a 0 + a + a n + α n The n th convergent of α, provided n N (if N exists), is given by c n = a 0 + a + a n + a n
2 If n N, then the n th convergent of α is simply c n = a 0 + a + a N + a N = c N The list [a 0, ] is the continued fraction representation of α Lemma The convergents are given by c n = pn where [ ] [ ] [ pn+ p n pn p = n an+ 0 or + p n+ = p n a n+ + p n + = a n+ + ] with p 0 = a 0, q 0 =, p = a 0 a +, q = a Note that for all i 0, q i > 0 In particular, q i is strictly monotonically increasing until a i+ = 0, or when i = N Proof First we claim that for any n and a formal variable Z we have p n Z + p n Z + = a 0 + a + a n + Z, () which we prove by induction The base case n = is an exercise Let n From the inductive hypothesis with Z a n+ + Z, a 0 + a + a n+ + Z, = p ( n an+ + ) Z + pn ( an+ + Z ) + qn = p n (a n+ Z + ) + p n Z (a n+ Z + ) + Z = p na n+ Z + p n + p n Z a n+ Z + + Z = p n+z + p n + Z + If we substitute Z = a n+ in (??), we get that c n+ = p n+ /+ as desired 2
3 Observation 2 Equation (??) is valid even if n > N, for the proof did not use n N Note that so p p det 0 q q 0 an+ = det 0 =, pn p det n = ( ) n This implies that p n, are relatively prime (because the ideal they generate in Z contains ), and so the convergent pn is already in lowest terms 2 Approximation Now we move to the approximation of real numbers by rational numbers Our aim is to use the lowest denominator rational number possible and still get a nice approximation How good a rational approximation can one get to a given real number α? One trivial rational approximation to α is a number a q with α a q 2q, (for any q we can simply choose a Z at a distance at most /2 from qα) The following theorem shows that every α has a pretty good rational approximation: Theorem 3 (Dirichlet s Theorem) For every α R, either α is rational, or else there are infinitely many distinct rational numbers a q such that α a q < Observation 4 We prove Theorem?? by proving the following stronger theorem: Theorem 5 For all Q N, there is a rational number a q α a q < qq Lemma 6 Theorem?? implies Theorem?? such that q Q and Proof of Lemma?? Take some Q N; by Theorem?? we can find a q α a q < q Q q 2 3 with
4 If a q = α, then α is rational, and so we are done with Theorem?? in this case If not, ie α a q > 0, then we can take a new Q2 so large that α a q > Q 2 Let a 2 be the pair guaranteed by Theorem?? for Q 2 Then α a 2 < α a q < Q 2 α a, so a 2 is distinct from a q Furthermore, α a 2 < Q 2 q2 2 q Continuing this process, we either find that α is rational, or else we find infinitely many a q required in Theorem?? as This proof, due to Dirichlet, was possibly the first appearance of the Dirichlet Pigeon Hole Principle Proof of Theorem?? If we show that there is some q with qα closer than Q to some integer, then we can use that integer as our a and be done Consider {0 mod, α mod, Qα mod } This is [ an Q ) + element set lying in the interval [0, ) There must be two in one of the Q intervals i Q, i Q, i {,, Q} Hence there exist r, s such that 0 sα mod rα mod = (s r)α mod < Q, viewed as an element of [0, ) Then there is an integer, namely that integer directly below (s r)α, such that (s r)α a < Q Our q is s r Let us now give an algorithm to construct these good approximations Concretely, the problem we want to solve is this: Problem Given a rational number α = r s and an integer Q, find a q possible subject to q Q with α a q as small as The input size for this problem is O(log r + log s + log Q) bits, and we will be interested in getting an algorithm which runs in time polynomial in this Our algorithm will be based on continued fractions (and we will use the notation we used in that section) It is not true that the solution to our rational approximation problem will be a convergent, but convergents will help us compute the solution easily We begin with a lemma on the sign of c n α Lemma 7 If n is even, c n α, else c n α Proof If n > N, this is trivial because c n = α, so we assume that n N We proceed by induction on n (showing the result for all α at the same time) 4
5 The lemma is clearly true for n = 0 (since c 0 = a 0 = α ) Now let n N, and assume the lemma holds for n for every α Then c n+ = a 0 + a + a n + a n+ = a 0 + c, n where c n is the n th convergent of α, the number such that α = a 0 + α c n is either at least (if n is odd) or at most α (if n is even) by our inductive assumption By similar reasoning to that showing α n > 0 for n > 0, c n > 0 Then a 0 + c is either at least α (if n + is even) or at most α n (if n + is odd) Now we characterize the best rational approximation with denominator Q Theorem 8 Let n be largest number such that Q, and let t be the greatest integer such that t + Q Then either pn is a solution to the rational approximation problem or tpn+p n t+ Proof Let L = tpn+p n t+ We will assume n is even The proof when n is odd will be very similar Because n is even, pn α and p n+ + α Note that if s = a n+, then spn+p n s+ = p n+ + ; if s = 0, then spn+p n s+ p n+ + This is because = p n As s ranges from 0 to a n+, spn+p n s+ d ds ( ) spn + p n = s + pn p det n ranges monotonically from p n (s + ) 2 ; does not change sign Hence, L must be between p n+ + and p n, both of which are at least α Hence L is at least α If r s is strictly closer to α than both c n and L, then r s I = (c n, L) For contradiction, suppose 0 < s Q Then r s p n, s because r s pn, and similarly Observe that I = I = pn tpn+p n r s L s(t + ) t+ = (t+ ) Then (t + ) s(t + ) + = ( + ) s s t + So s (t + ) + > Q, a contradiction = s ( ) tqn + + (t + ) Observation 9 We saw in the proof that if n is even, α (c n, L), and if n is odd, α (L, c n ) Further, the interval has length exactly (t+ ) Then c n α (t+ ) 5 to
6 As we noted before, increases monotonically until n = N As a b is the best approximation with denominator at most Q for any Q b, we cannot have > b for any n If so, then + > b as well By Theorem??, either L or p n+ + would at least as good an approximation as a b with denominator at most + For any t > 0, t+ + > +, so t = 0 Then L = pn This is impossible, because p n qn and p n+ + are reduced and therefore neither are equal to a b This also applies that N exists for α if and only if α is rational, ie the continued fraction representation of α terminates iff α is rational 3 Running Time It will be of use to bound p n and We already know that b From Remark??, we know that a p n b Hence p n qn b a + = O(a) We know that the n from Theorem?? is O(log(Q)) because the grow at least as fast as the n th entry of the Fibonacci sequence provided n N At each step in the computation of the continued fraction, we compute the floor of a rational number α i = r i s i, which is simply the number of times the denominator of α i goes into the numerator This is division with remainder, which we know to take O(log(r i ) log(s i )) operations We know α i = α i α i, which has denominator the remainder of r i under division by s i (which is strictly smaller than s i ) and numerator s i As α 0 = α = a b, we can conclude by induction that r i, s i max(a, b) and so the number of operations per step is certainly at most O((log(a) + log(b)) 2 ) Finding t can take at most O(log(Q) 2 ) steps, as we find it by division with remainder of Q by t is at most a n+, which is at most + b p n, p n = O(a) Taken together, finding L must take O(poly(log(a), log(b), log(q))) Finding whether L or c n is closer to α can also certainly take at most O(poly(log(a), log(b), log(q))), and finding the continued fraction is also O(poly(log(a), log(b), log(q))) Hence the entire running time must be O(poly(log(a), log(b), log(q))) 6
Mathematics 228(Q1), Assignment 2 Solutions
Mathematics 228(Q1), Assignment 2 Solutions Exercise 1.(10 marks) A natural number n > 1 is said to be square free if d N with d 2 n implies d = 1. Show that n is square free if and only if n = p 1 p k
More informationNotes on Continued Fractions for Math 4400
. Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural
More informationMATH10040: Numbers and Functions Homework 1: Solutions
MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3
More informationCONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS
CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas
More informationREAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental
More informationMath 109 HW 9 Solutions
Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we
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 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 informationPigeonhole Principle and Ramsey Theory
Pigeonhole Principle and Ramsey Theory The Pigeonhole Principle (PP) has often been termed as one of the most fundamental principles in combinatorics. The familiar statement is that if we have n pigeonholes
More informationcse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska
cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska LECTURE 12 CHAPTER 4 NUMBER THEORY PART1: Divisibility PART 2: Primes PART 1: DIVISIBILITY Basic Definitions Definition Given m,n Z, we say
More informationINTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes.
INTEGERS PETER MAYR (MATH 2001, CU BOULDER) In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. 1. Divisibility Definition. Let a, b
More informationmeans is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.
1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for
More informationThe following techniques for methods of proofs are discussed in our text: - Vacuous proof - Trivial proof
Ch. 1.6 Introduction to Proofs The following techniques for methods of proofs are discussed in our text - Vacuous proof - Trivial proof - Direct proof - Indirect proof (our book calls this by contraposition)
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationLecture 6: Deterministic Primality Testing
Lecture 6: Deterministic Primality Testing Topics in Pseudorandomness and Complexity (Spring 018) Rutgers University Swastik Kopparty Scribe: Justin Semonsen, Nikolas Melissaris 1 Introduction The AKS
More informationIS 709/809: Computational Methods for IS Research. Math Review: Algorithm Analysis
IS 709/809: Computational Methods for IS Research Math Review: Algorithm Analysis Nirmalya Roy Department of Information Systems University of Maryland Baltimore County www.umbc.edu Topics Proof techniques
More informationHMMT February 2018 February 10, 2018
HMMT February 018 February 10, 018 Algebra and Number Theory 1. For some real number c, the graphs of the equation y = x 0 + x + 18 and the line y = x + c intersect at exactly one point. What is c? 18
More informationSolutions to Assignment 1
Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive
More informationSolution Set 2. Problem 1. [a] + [b] = [a + b] = [b + a] = [b] + [a] ([a] + [b]) + [c] = [a + b] + [c] = [a + b + c] = [a] + [b + c] = [a] + ([b + c])
Solution Set Problem 1 (1) Z/nZ is the set of equivalence classes of Z mod n. Equivalence is determined by the following rule: [a] = [b] if and only if b a = k n for some k Z. The operations + and are
More informationExercises Exercises. 2. Determine whether each of these integers is prime. a) 21. b) 29. c) 71. d) 97. e) 111. f) 143. a) 19. b) 27. c) 93.
Exercises Exercises 1. Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143 2. Determine whether each of these integers is prime. a) 19 b) 27 c) 93 d) 101 e) 107 f)
More informationLecture 3.2: Induction, and Strong Induction
Lecture 3.2: Induction, and Strong Induction CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag Course Admin Mid-Term 1 Graded; to be returned
More informationTHE DIVISION THEOREM IN Z AND R[T ]
THE DIVISION THEOREM IN Z AND R[T ] KEITH CONRAD 1. Introduction In both Z and R[T ], we can carry out a process of division with remainder. Theorem 1.1. For any integers a and b, with b nonzero, there
More informationP-adic numbers. Rich Schwartz. October 24, 2014
P-adic numbers Rich Schwartz October 24, 2014 1 The Arithmetic of Remainders In class we have talked a fair amount about doing arithmetic with remainders and now I m going to explain what it means in a
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 information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationMath 110 HW 3 solutions
Math 0 HW 3 solutions May 8, 203. For any positive real number r, prove that x r = O(e x ) as functions of x. Suppose r
More informationWe have been going places in the car of calculus for years, but this analysis course is about how the car actually works.
Analysis I We have been going places in the car of calculus for years, but this analysis course is about how the car actually works. Copier s Message These notes may contain errors. In fact, they almost
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 informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
More informationLecture 11 - Basic Number Theory.
Lecture 11 - Basic Number Theory. Boaz Barak October 20, 2005 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that a divides b,
More informationDR.RUPNATHJI( DR.RUPAK NATH )
Contents 1 Sets 1 2 The Real Numbers 9 3 Sequences 29 4 Series 59 5 Functions 81 6 Power Series 105 7 The elementary functions 111 Chapter 1 Sets It is very convenient to introduce some notation and terminology
More informationSums of Squares. Bianca Homberg and Minna Liu
Sums of Squares Bianca Homberg and Minna Liu June 24, 2010 Abstract For our exploration topic, we researched the sums of squares. Certain properties of numbers that can be written as the sum of two squares
More informationInference and Proofs (1.6 & 1.7)
EECS 203 Spring 2016 Lecture 4 Page 1 of 9 Introductory problem: Inference and Proofs (1.6 & 1.7) As is commonly the case in mathematics, it is often best to start with some definitions. An argument for
More informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
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 informationMath1a Set 1 Solutions
Math1a Set 1 Solutions October 15, 2018 Problem 1. (a) For all x, y, z Z we have (i) x x since x x = 0 is a multiple of 7. (ii) If x y then there is a k Z such that x y = 7k. So, y x = (x y) = 7k is also
More information3 Finite continued fractions
MTH628 Number Theory Notes 3 Spring 209 3 Finite continued fractions 3. Introduction Let us return to the calculation of gcd(225, 57) from the preceding chapter. 225 = 57 + 68 57 = 68 2 + 2 68 = 2 3 +
More informationPRACTICE PROBLEMS: SET 1
PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if
More informationSeunghee Ye Ma 8: Week 2 Oct 6
Week 2 Summary This week, we will learn about sequences and real numbers. We first define what we mean by a sequence and discuss several properties of sequences. Then, we will talk about what it means
More informationMATH 55 - HOMEWORK 6 SOLUTIONS. 1. Section = 1 = (n + 1) 3 = 2. + (n + 1) 3. + (n + 1) 3 = n2 (n + 1) 2.
MATH 55 - HOMEWORK 6 SOLUTIONS Exercise Section 5 Proof (a) P () is the statement ( ) 3 (b) P () is true since ( ) 3 (c) The inductive hypothesis is P (n): ( ) n(n + ) 3 + 3 + + n 3 (d) Assuming the inductive
More informationD-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma.
D-MATH Algebra I HS18 Prof. Rahul Pandharipande Solution 1 Arithmetic, Zorn s Lemma. 1. (a) Using the Euclidean division, determine gcd(160, 399). (b) Find m 0, n 0 Z such that gcd(160, 399) = 160m 0 +
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers Copyright
More informationNORTHWESTERN UNIVERSITY Thrusday, Oct 6th, 2011 ANSWERS FALL 2011 NU PUTNAM SELECTION TEST
Problem A1. Let a 1, a 2,..., a n be n not necessarily distinct integers. exist a subset of these numbers whose sum is divisible by n. Prove that there - Answer: Consider the numbers s 1 = a 1, s 2 = a
More informationbe any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism
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 UFD. Therefore
More informationBeautiful Mathematics
Beautiful Mathematics 1. Principle of Mathematical Induction The set of natural numbers is the set of positive integers {1, 2, 3,... } and is denoted by N. The Principle of Mathematical Induction is a
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationCHAPTER 1 REAL NUMBERS KEY POINTS
CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationMath /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs
Math 4030-001/Foundations of Algebra/Fall 017 Foundations of the Foundations: Proofs A proof is a demonstration of the truth of a mathematical statement. We already know what a mathematical statement is.
More informationContest Number Theory
Contest Number Theory Andre Kessler December 7, 2008 Introduction Number theory is one of the core subject areas of mathematics. It can be somewhat loosely defined as the study of the integers. Unfortunately,
More informationPutnam Greedy Algorithms Cody Johnson. Greedy Algorithms. May 30, 2016 Cody Johnson.
1 Introduction Greedy Algorithms May 0, 2016 Cody Johnson ctj@math.cmu.edu A greedy algorithm is an algorithm that chooses the optimal choice in the short run. 2 Examples 1. Prove that every nonnegative
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
More informationFinite Fields: An introduction through exercises Jonathan Buss Spring 2014
Finite Fields: An introduction through exercises Jonathan Buss Spring 2014 A typical course in abstract algebra starts with groups, and then moves on to rings, vector spaces, fields, etc. This sequence
More informationTable of Contents. 2013, Pearson Education, Inc.
Table of Contents Chapter 1 What is Number Theory? 1 Chapter Pythagorean Triples 5 Chapter 3 Pythagorean Triples and the Unit Circle 11 Chapter 4 Sums of Higher Powers and Fermat s Last Theorem 16 Chapter
More informationCHAPTER 3. Number Theory
CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss (1777-1855) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated
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 informationCorollary 4.2 (Pepin s Test, 1877). Let F k = 2 2k + 1, the kth Fermat number, where k 1. Then F k is prime iff 3 F k 1
4. Primality testing 4.1. Introduction. Factorisation is concerned with the problem of developing efficient algorithms to express a given positive integer n > 1 as a product of powers of distinct primes.
More informationMATH3283W LECTURE NOTES: WEEK 6 = 5 13, = 2 5, 1 13
MATH383W LECTURE NOTES: WEEK 6 //00 Recursive sequences (cont.) Examples: () a =, a n+ = 3 a n. The first few terms are,,, 5 = 5, 3 5 = 5 3, Since 5
More informationLecture Overview. 2 Weak Induction
COMPSCI 30: Discrete Mathematics for Computer Science February 18, 019 Lecturer: Debmalya Panigrahi Lecture 11 Scribe: Kevin Sun 1 Overview In this lecture, we study mathematical induction, which we often
More informationProof Techniques (Review of Math 271)
Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil
More informationMATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology.
MATH 501 Discrete Mathematics Lecture 6: Number theory Prof. Dr. Slim Abdennadher, slim.abdennadher@guc.edu.eg German University Cairo, Department of Media Engineering and Technology 1 Number theory Number
More informationFACTORIZATION OF IDEALS
FACTORIZATION OF IDEALS 1. General strategy Recall the statement of unique factorization of ideals in Dedekind domains: Theorem 1.1. Let A be a Dedekind domain and I a nonzero ideal of A. Then there are
More informationDiscrete Mathematics and Probability Theory Fall 2013 Vazirani Note 1
CS 70 Discrete Mathematics and Probability Theory Fall 013 Vazirani Note 1 Induction Induction is a basic, powerful and widely used proof technique. It is one of the most common techniques for analyzing
More informationMATH 2400 LECTURE NOTES: POLYNOMIAL AND RATIONAL FUNCTIONS. Contents 1. Polynomial Functions 1 2. Rational Functions 6
MATH 2400 LECTURE NOTES: POLYNOMIAL AND RATIONAL FUNCTIONS PETE L. CLARK Contents 1. Polynomial Functions 1 2. Rational Functions 6 1. Polynomial Functions Using the basic operations of addition, subtraction,
More informationQuantitative Aptitude
WWW.UPSCMANTRA.COM Quantitative Aptitude Concept 1 1. Number System 2. HCF and LCM 2011 Prelims Paper II NUMBER SYSTEM 2 NUMBER SYSTEM In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7,
More informationMathematical Induction
Mathematical Induction Let s motivate our discussion by considering an example first. What happens when we add the first n positive odd integers? The table below shows what results for the first few values
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 informationDivisibility in the Fibonacci Numbers. Stefan Erickson Colorado College January 27, 2006
Divisibility in the Fibonacci Numbers Stefan Erickson Colorado College January 27, 2006 Fibonacci Numbers F n+2 = F n+1 + F n n 1 2 3 4 6 7 8 9 10 11 12 F n 1 1 2 3 8 13 21 34 89 144 n 13 14 1 16 17 18
More informationCSC 344 Algorithms and Complexity. Proof by Mathematical Induction
CSC 344 Algorithms and Complexity Lecture #1 Review of Mathematical Induction Proof by Mathematical Induction Many results in mathematics are claimed true for every positive integer. Any of these results
More informationInstructor: Bobby Kleinberg Lecture Notes, 25 April The Miller-Rabin Randomized Primality Test
Introduction to Algorithms (CS 482) Cornell University Instructor: Bobby Kleinberg Lecture Notes, 25 April 2008 The Miller-Rabin Randomized Primality Test 1 Introduction Primality testing is an important
More informationNumber Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z:
Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some z Z Notation: b Fact: for all, b, c Z:, 1, and 0 0 = 0 b and b c = c b and c = (b + c) b and b = ±b 1
More informationSect Introduction to Rational Expressions
127 Sect 7.1 - Introduction to Rational Expressions Concept #1 Definition of a Rational Expression. Recall that a rational number is any number that can be written as the ratio of two integers where the
More information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
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 informationMathematical Induction. Part Two
Mathematical Induction Part Two The principle of mathematical induction states that if for some property P(n), we have that If it starts P(0) is true and and it keeps going For any n N, we have P(n) P(n
More informationProof worksheet solutions
Proof worksheet solutions These are brief, sketched solutions. Comments in blue can be ignored, but they provide further explanation and outline common misconceptions Question 1 (a) x 2 + 4x +12 = (x +
More informationIntroduction to Basic Proof Techniques Mathew A. Johnson
Introduction to Basic Proof Techniques Mathew A. Johnson Throughout this class, you will be asked to rigorously prove various mathematical statements. Since there is no prerequisite of a formal proof class,
More informationCHAPTER 6. Prime Numbers. Definition and Fundamental Results
CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationFermat's Little Theorem
Fermat's Little Theorem CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Not to be confused with... Fermat's Last Theorem: x n + y n = z n has no integer solution for n > 2 Recap: Modular Arithmetic
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationSolutions for Homework Assignment 2
Solutions for Homework Assignment 2 Problem 1. If a,b R, then a+b a + b. This fact is called the Triangle Inequality. By using the Triangle Inequality, prove that a b a b for all a,b R. Solution. To prove
More informationInfinite Continued Fractions
Infinite Continued Fractions 8-5-200 The value of an infinite continued fraction [a 0 ; a, a 2, ] is lim c k, where c k is the k-th convergent k If [a 0 ; a, a 2, ] is an infinite continued fraction with
More informationLecture 11: Cantor-Zassenhaus Algorithm
CS681 Computational Number Theory Lecture 11: Cantor-Zassenhaus Algorithm Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview In this class, we shall look at the Cantor-Zassenhaus randomized
More informationSolutions to Practice Final
s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (
More informationCS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II
CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2
More informationMATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions
MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a
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 informationComplete Induction and the Well- Ordering Principle
Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k
More informationRational Approximation by Continued Fractions
Rational Approximation by Continued Fractions The convergents of a continued fraction expansion of x give the best rational approximations to x. Specifically, the only way a fraction can approximate x
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationInduction. Induction. Induction. Induction. Induction. Induction 2/22/2018
The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers. It cannot be used to discover theorems, but only to prove them. If we have
More informationThis is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.
8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only
More informationAn integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.
Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1
More informationRamsey Theory. May 24, 2015
Ramsey Theory May 24, 2015 1 König s Lemma König s Lemma is a basic tool to move between finite and infinite combinatorics. To be concise, we use the notation [k] = {1, 2,..., k}, and [X] r will denote
More informationThe standard form for a general polynomial of degree n is written. Examples of a polynomial in standard form
Section 4 1A: The Rational Zeros (Roots) of a Polynomial The standard form for a general polynomial of degree n is written f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 where the highest degree term
More informationThe Real Number System
MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely
More information