. In particular if a b then N(
|
|
- Adele Powers
- 6 years ago
- Views:
Transcription
1 Gaussian Integers II Let us summarise what we now about Gaussian integers so far: If a, b Z[ i], then N( ab) N( a) N( b). In particular if a b then N( a ) N( b). Let z Z[i]. If N( z ) is an integer prime, then z is a Gaussian prime. A prime p Z is a Gaussian prime if and only if p y has no solution. Fundamental Theorem of Arithmetic Unique prime factorization Euclidean Division Theorem, Algorithm, and GCD algorithm GCD property: the gcd of two Gaussian integers is a linear combination of them. Prime Divisor Property If p is a Gaussian prime and p wz then p w or p z. We are going to use these properties to prove the following theorem: Fermat s Two Squares Theorem Let p be a prime. The equation p mod 4. y has a solution if and only if p = or p 1 We already now that y p has no solution when p 3 mod 4, by Theorem 3.30 in the boo: if y p then y 0 mod p, which by Theorem 3.30 implies 0 y mod p, and so and y are multiples of p. But it is then impossible that their squares add up to p in the integers. The main part of the two squares theorem is to show that if p 1 mod 4 then that equation can be solved. The ey is to figure out eactly which Gaussian integers are primes. 4. Fermat s Two Squares Theorem We already now that a composite integer cannot be a Gaussian prime. By Theorem.1, the primes p that are Gaussian primes are those with no solution to p y has no solution. Lemma.1 says that any Gaussian integer of prime norm is a Gaussian prime. We show these are the only other Gaussian primes: Theorem 4.1. The Gaussian primes are units times integer primes not of the form Gaussian integers of prime norm. y or
2 Proof. Let z denote the comple conjugate of a Gaussian integer z, so N ( zz. We have to show that if z is a Gaussian prime, then z is an integer prime or N ( is an integer prime. Suppose z is a Gaussian prime, but N ( is not an integer prime. Then N ( mn for some positive integers m > 1 and n > 1. Since z is a Gaussian prime, so is z [Eercise]. By the unique factorization theorem in the Gaussian integers, the unique prime factorization of N ( must be N ( zz. But N ( mn is another factorization, so m and n must be the Gaussian primes z and z up to multiplication by a unit. This means m and n must be an integer prime and its comple conjugate. But m > 1 and n > 1, so the only possibility is m n p for some integer prime p, and then z is p or a unit times p, as required. Lagrange s Lemma. If p 1 mod 4 is prime, then p n 1 for some integer n. Proof. Write p 4m 1. By Wilson s Theorem, and m i (m i 1) mod p, (4m)! (m )! 1 mod p. Putting n ( m)! we habe n 1 0 mod p. Fermat s Two Squares Theorem Let p be a prime. The equation p mod 4. y has a solution if and only if p = or p 1 Proof. Theorem 3.30 says if p y then p is not 3 mod 4 i.e. p = or p 1 mod 4. For the reverse implication, it is trivial for p = so we concentrate on p 1 mod 4. By Lagrange s Lemma, p n 1 for some integer n. Now n 1 ( n i )( n i ) and so by the prime divisor property, if p is a Gaussian prime, then p n i or p n i. These are both impossible [Eercise], so p can t be a Gaussian prime. By Theorem.1, that means p y has a solution. In fact one can show that there is only one solution to p y with y.
3 5. Pythagorean Triples We return to Diophantine equations. Specifically, let us study equations of the form y z where > 1 is an integer. This includes the eample of Pythagorean triples y z. At the beginning of the notes, we outlined the strategy: 1. Factor y in the Gaussian integers: y ( (.. Show that iy, iy are relatively prime 3. Use prime divisor property: (,( are both units times th powers. 4. Solve iy ( u iv) The third step is justified by the following lemma. Lemma 5.1 Let, y and z be Gaussian integers such that y z and suppose and y are relatively prime. Then for some unit u, and some Gaussian integers a and b, a and y b u. e e e Proof. If z z z r 1 1 r is the prime factorization of z, where the z i are distinct Gaussian e e e primes, the prime factorization of y is z z z r 1 1 r. By the prime divisor property, e z i e i y implies z i e i or z i i y for i = 1,...,r. If the first s of them divide, then we have z e1 e e 1 z z s s e i i ( z ) and similarly for y. The second step depends more on the equation. It is not true in general that a Gaussian integer and its comple conjugate are relatively prime, for instance i, i are not relatively prime. Lemma 5. Suppose y z for integers, y and z, where (, y, 1. Then iy, iy are relatively prime.
4 Proof. Let w be a GCD of iy, iy. We have to show w is a unit. We now that w ( w ( ( ( w iy w Since (, y, 1 and y z we have (, y) 1. So that means w. The prime factorization of is (1 i)(1 i) so w is a unit times 1 i or a unit times 1 i or a unit. But if w is a unit times 1 i or 1 i we have a problem: w y (1 i ) z i z z 4 z and mod 4 we have y 0. This implies 0 y mod 4, but that is impossible since (, y) 1. We conclude that w is a unit. With these two lemmas, we can now solve the Diophantine equation. Theorem 5.1 The positive integer solutions to of triples (, y, such that (, y, 1 and y z are given by all positive integer multiples y v or y v z z where u, ( u, v) 1, and eactly one of u and v is even. Proof. We conclude from Lemma 5.1 and 5. that there eists a unit t such that ( i y) ta,( b t for some Gaussian integers a and b. Suppose a U iv. Then the first equation gives ( i y) t( U iv) t( U V ) tiuv. Now t { 1, 1, i, i}. We consider each case.
5 If t = 1, then U V, y UV and we let u U, v V If t = -1, then V U, y UV and we let u V, v U If t = i, then y U V, UV and we let u U, v V If t = -i, then y V U, UV and we let u V, v U In all cases, we have acquired solutions of the required form and u > v since we are looing only at positive solutions. It remains to show eactly one of u and v is even, and (u,v) = 1. If u and v have a common factor, then it is also a common factor of and y and z, which contradicts (, y, 1. So (u,v) = 1. In particular, they can t both be even. If they are both odd, then, y and z would all be even, another contradiction to (, y, 1. So eactly one of u and v is even. Eample. Solve prime. By Lemma 5.1, 3 1 z. Well by Lemma 5. with y = 1, i, i are relatively i 3 3 ta, i b t for some unit t and some Gaussian integers imaginary parts we get a iv and b. If t 1, comparing 3 1 Im( u i v) (3u ). If t i comparing imaginary parts we get 1 Im i ( u iv) ( u 3v ). The only possibilities are ( u, v) (1,0 ) and ( u, v) (0,1 ). Both cases give 3 Re( u iv ) 3 0 and so the only solutions are 0, z 1.
6
Hilbert s theorem 90, Dirichlet s unit theorem and Diophantine equations
Hilbert s theorem 90, Dirichlet s unit theorem and Diophantine equations B. Sury Stat-Math Unit Indian Statistical Institute 8th Mile Mysore Road Bangalore - 560 059 India. sury@isibang.ac.in Introduction
More informationMath Theory of Number Homework 1
Math 4050 Theory of Number Homework 1 Due Wednesday, 015-09-09, in class Do 5 of the following 7 problems. Please only attempt 5 because I will only grade 5. 1. Find all rational numbers and y satisfying
More informationGaussian integers. 1 = a 2 + b 2 = c 2 + d 2.
Gaussian integers 1 Units in Z[i] An element x = a + bi Z[i], a, b Z is a unit if there exists y = c + di Z[i] such that xy = 1. This implies 1 = x 2 y 2 = (a 2 + b 2 )(c 2 + d 2 ) But a 2, b 2, c 2, d
More informationSolving Diophantine Equations With Unique Factorization
Solving Diophantine Equations With Unique Factorization February 17, 2016 1 Introduction In this note we should how unique factorization in rings like Z[i] and Z[ 2] can be used to find integer solutions
More informationMath 412: Number Theory Lecture 26 Gaussian Integers II
Math 412: Number Theory Lecture 26 Gaussian Integers II Gexin Yu gyu@wm.edu College of William and Mary Let i = 1. Complex numbers of the form a + bi with a, b Z are called Gaussian integers. Let z = a
More informationa the relation arb is defined if and only if = 2 k, k
DISCRETE MATHEMATICS Past Paper Questions in Number Theory 1. Prove that 3k + 2 and 5k + 3, k are relatively prime. (Total 6 marks) 2. (a) Given that the integers m and n are such that 3 (m 2 + n 2 ),
More informationWilson s Theorem and Fermat s Little Theorem
Wilson s Theorem and Fermat s Little Theorem Wilson stheorem THEOREM 1 (Wilson s Theorem): (p 1)! 1 (mod p) if and only if p is prime. EXAMPLE: We have (2 1)!+1 = 2 (3 1)!+1 = 3 (4 1)!+1 = 7 (5 1)!+1 =
More informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More informationProof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have
Exercise 13. Consider positive integers a, b, and c. (a) Suppose gcd(a, b) = 1. (i) Show that if a divides the product bc, then a must divide c. I give two proofs here, to illustrate the different methods.
More informationChuck Garner, Ph.D. May 25, 2009 / Georgia ARML Practice
Some Chuck, Ph.D. Department of Mathematics Rockdale Magnet School for Science Technology May 25, 2009 / Georgia ARML Practice Outline 1 2 3 4 Outline 1 2 3 4 Warm-Up Problem Problem Find all positive
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 informationNumber Theory. Final Exam from Spring Solutions
Number Theory. Final Exam from Spring 2013. Solutions 1. (a) (5 pts) Let d be a positive integer which is not a perfect square. Prove that Pell s equation x 2 dy 2 = 1 has a solution (x, y) with x > 0,
More informationMathematics of Cryptography Part I
CHAPTER 2 Mathematics of Crptograph Part I (Solution to Practice Set) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinit to positive infinit. The set of
More informationNotes: Pythagorean Triples
Math 5330 Spring 2018 Notes: Pythagorean Triples Many people know that 3 2 + 4 2 = 5 2. Less commonly known are 5 2 + 12 2 = 13 2 and 7 2 + 24 2 = 25 2. Such a set of integers is called a Pythagorean Triple.
More informationAll variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.
Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1
More informationCOMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635
COMP239: Mathematics for Computer Science II Prof. Chadi Assi assi@ciise.concordia.ca EV7.635 The Euclidean Algorithm The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is
More informationEXAMPLES OF MORDELL S EQUATION
EXAMPLES OF MORDELL S EQUATION KEITH CONRAD 1. Introduction The equation y 2 = x 3 +k, for k Z, is called Mordell s equation 1 on account of Mordell s long interest in it throughout his life. A natural
More information6]. (10) (i) Determine the units in the rings Z[i] and Z[ 10]. If n is a squarefree
Quadratic extensions Definition: Let R, S be commutative rings, R S. An extension of rings R S is said to be quadratic there is α S \R and monic polynomial f(x) R[x] of degree such that f(α) = 0 and S
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationTry the assignment f(1) = 2; f(2) = 1; f(3) = 4; f(4) = 3;.
I. Precisely complete the following definitions: 1. A natural number n is composite whenever... See class notes for the precise definitions 2. Fix n in N. The number s(n) represents... 3. For A and B sets,
More informationTHE TRIANGULAR THEOREM OF THE PRIMES : BINARY QUADRATIC FORMS AND PRIMITIVE PYTHAGOREAN TRIPLES
THE TRIANGULAR THEOREM OF THE PRIMES : BINARY QUADRATIC FORMS AND PRIMITIVE PYTHAGOREAN TRIPLES Abstract. This article reports the occurrence of binary quadratic forms in primitive Pythagorean triangles
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 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 informationSchool of Mathematics
School of Mathematics Programmes in the School of Mathematics Programmes including Mathematics Final Examination Final Examination 06 22498 MSM3P05 Level H Number Theory 06 16214 MSM4P05 Level M Number
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 informationChapter 5.1: Induction
Chapter.1: Induction Monday, July 1 Fermat s Little Theorem Evaluate the following: 1. 1 (mod ) 1 ( ) 1 1 (mod ). (mod 7) ( ) 8 ) 1 8 1 (mod ). 77 (mod 19). 18 (mod 1) 77 ( 18 ) 1 1 (mod 19) 18 1 (mod
More informationProposition The gaussian numbers form a field. The gaussian integers form a commutative ring.
Chapter 11 Gaussian Integers 11.1 Gaussian Numbers Definition 11.1. A gaussian number is a number of the form z = x + iy (x, y Q). If x, y Z we say that z is a gaussian integer. Proposition 11.1. The gaussian
More informationFavorite Topics from Complex Arithmetic, Analysis and Related Algebra
Favorite Topics from Complex Arithmetic, Analysis and Related Algebra construction at 09FALL/complex.tex Franz Rothe Department of Mathematics University of North Carolina at Charlotte Charlotte, NC 3
More informationNumber Theory. CSS322: Security and Cryptography. Sirindhorn International Institute of Technology Thammasat University CSS322. Number Theory.
CSS322: Security and Cryptography Sirindhorn International Institute of Technology Thammasat University Prepared by Steven Gordon on 29 December 2011 CSS322Y11S2L06, Steve/Courses/2011/S2/CSS322/Lectures/number.tex,
More informationExplicit Methods in Algebraic Number Theory
Explicit Methods in Algebraic Number Theory Amalia Pizarro Madariaga Instituto de Matemáticas Universidad de Valparaíso, Chile amaliapizarro@uvcl 1 Lecture 1 11 Number fields and ring of integers Algebraic
More informationExploring Number Theory via Diophantine Equations
Exploring Number Theory via Diophantine Equations Department of Mathematics Colorado College Fall, 2009 Outline Some History Linear Pythagorean Triples Introduction to Continued Fractions Elementary Problems
More informationMATH 152 Problem set 6 solutions
MATH 52 Problem set 6 solutions. Z[ 2] is a Euclidean domain (i.e. has a division algorithm): the idea is to approximate the quotient by an element in Z[ 2]. More precisely, let a+b 2, c+d 2 Z[ 2] (of
More informationSome Highlights along a Path to Elliptic Curves
11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational
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 informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationCh 4.2 Divisibility Properties
Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:
More information20th Bay Area Mathematical Olympiad. BAMO 2018 Problems and Solutions. February 27, 2018
20th Bay Area Mathematical Olympiad BAMO 201 Problems and Solutions The problems from BAMO- are A E, and the problems from BAMO-12 are 1 5. February 27, 201 A Twenty-five people of different heights stand
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 informationProof by Contradiction
Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving
More informationPrimes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1
Primes Rational, Gaussian, Industrial Strength, etc Robert Campbell 11/29/2010 1 Primes and Theory Number Theory to Abstract Algebra History Euclid to Wiles Computation pencil to supercomputer Practical
More informationMath 324, Fall 2011 Assignment 7 Solutions. 1 (ab) γ = a γ b γ mod n.
Math 324, Fall 2011 Assignment 7 Solutions Exercise 1. (a) Suppose a and b are both relatively prime to the positive integer n. If gcd(ord n a, ord n b) = 1, show ord n (ab) = ord n a ord n b. (b) Let
More informationProofs. Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm. Reading (Epp s textbook)
Proofs Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm Reading (Epp s textbook) 4.3 4.8 1 Divisibility The notation d n is read d divides n. Symbolically,
More informationCourse 2316 Sample Paper 1
Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity
More information2.3 In modular arithmetic, all arithmetic operations are performed modulo some integer.
CHAPTER 2 INTRODUCTION TO NUMBER THEORY ANSWERS TO QUESTIONS 2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m are integers. That is, b is a divisor of a if there is no remainder
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 informationPREPARATION NOTES FOR NUMBER THEORY PRACTICE WED. OCT. 3,2012
PREPARATION NOTES FOR NUMBER THEORY PRACTICE WED. OCT. 3,2012 0.1. Basic Num. Th. Techniques/Theorems/Terms. Modular arithmetic, Chinese Remainder Theorem, Little Fermat, Euler, Wilson, totient, Euclidean
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 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 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 informationNumber Theory Solutions Packet
Number Theory Solutions Pacet 1 There exist two distinct positive integers, both of which are divisors of 10 10, with sum equal to 157 What are they? Solution Suppose 157 = x + y for x and y divisors of
More informationSignature: (In Ink) UNIVERSITY OF MANITOBA TEST 1 SOLUTIONS COURSE: MATH 2170 DATE & TIME: February 11, 2019, 16:30 17:15
PAGE: 1 of 7 I understand that cheating is a serious offence: Signature: (In Ink) PAGE: 2 of 7 1. Let a, b, m, be integers, m > 1. [1] (a) Define a b. Solution: a b iff for some d, ad = b. [1] (b) Define
More informationASSIGNMENT Use mathematical induction to show that the sum of the cubes of three consecutive non-negative integers is divisible by 9.
ASSIGNMENT 1 1. Use mathematical induction to show that the sum of the cubes of three consecutive non-negative integers is divisible by 9. 2. (i) If d a and d b, prove that d (a + b). (ii) More generally,
More information12x + 18y = 50. 2x + v = 12. (x, v) = (6 + k, 2k), k Z.
Math 3, Fall 010 Assignment 3 Solutions Exercise 1. Find all the integral solutions of the following linear diophantine equations. Be sure to justify your answers. (i) 3x + y = 7. (ii) 1x + 18y = 50. (iii)
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More informationA Readable Introduction to Real Mathematics
Solutions to selected problems in the book A Readable Introduction to Real Mathematics D. Rosenthal, D. Rosenthal, P. Rosenthal Chapter 7: The Euclidean Algorithm and Applications 1. Find the greatest
More informationARITHMETIC PROGRESSIONS IN CYCLES OF QUADRATIC POLYNOMIALS
ARITHMETIC PROGRESSIONS IN CYCLES OF QUADRATIC POLYNOMIALS TIMO ERKAMA It is an open question whether n-cycles of complex quadratic polynomials can be contained in the field Q(i) of complex rational numbers
More informationEXAMPLES OF MORDELL S EQUATION
EXAMPLES OF MORDELL S EQUATION KEITH CONRAD 1. Introduction The equation y 2 = x 3 +k, for k Z, is called Mordell s equation 1 on account of Mordell s long interest in it throughout his life. A natural
More informationNOTES ON INTEGERS. 1. Integers
NOTES ON INTEGERS STEVEN DALE CUTKOSKY The integers 1. Integers Z = {, 3, 2, 1, 0, 1, 2, 3, } have addition and multiplication which satisfy familar rules. They are ordered (m < n if m is less than n).
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationRings and modular arithmetic
Chapter 8 Rings and modular arithmetic So far, we have been working with just one operation at a time. But standard number systems, such as Z, have two operations + and which interact. It is useful to
More informationMATH 145 Algebra, Solutions to Assignment 4
MATH 145 Algebra, Solutions to Assignment 4 1: a) Find the inverse of 178 in Z 365. Solution: We find s and t so that 178s + 365t = 1, and then 178 1 = s. The Euclidean Algorithm gives 365 = 178 + 9 178
More informationPRIME LABELING OF SMALL TREES WITH GAUSSIAN INTEGERS. 1. Introduction
PRIME LABELING OF SMALL TREES WITH GAUSSIAN INTEGERS HUNTER LEHMANN AND ANDREW PARK Abstract. A graph on n vertices is said to admit a prime labeling if we can label its vertices with the first n natural
More information4 PRIMITIVE ROOTS Order and Primitive Roots The Index Existence of primitive roots for prime modulus...
PREFACE These notes have been prepared by Dr Mike Canfell (with minor changes and extensions by Dr Gerd Schmalz) for use by the external students in the unit PMTH 338 Number Theory. This booklet covers
More informationp = This is small enough that its primality is easily verified by trial division. A candidate prime above 1000 p of the form p U + 1 is
LARGE PRIME NUMBERS 1. Fermat Pseudoprimes Fermat s Little Theorem states that for any positive integer n, if n is prime then b n % n = b for b = 1,..., n 1. In the other direction, all we can say is that
More informationElementary Number Theory MARUCO. Summer, 2018
Elementary Number Theory MARUCO Summer, 2018 Problem Set #0 axiom, theorem, proof, Z, N. Axioms Make a list of axioms for the integers. Does your list adequately describe them? Can you make this list as
More information1. Algebra 1.7. Prime numbers
1. ALGEBRA 30 1. Algebra 1.7. Prime numbers Definition Let n Z, with n 2. If n is not a prime number, then n is called a composite number. We look for a way to test if a given positive integer is prime
More informationMATH342 Practice Exam
MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice
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 informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More informationa 2 + b 2 = (p 2 q 2 ) 2 + 4p 2 q 2 = (p 2 + q 2 ) 2 = c 2,
5.3. Pythagorean triples Definition. A Pythagorean triple is a set (a, b, c) of three integers such that (in order) a 2 + b 2 c 2. We may as well suppose that all of a, b, c are non-zero, and positive.
More informationROTH S THEOREM THE FOURIER ANALYTIC APPROACH NEIL LYALL
ROTH S THEOREM THE FOURIER AALYTIC APPROACH EIL LYALL Roth s Theorem. Let δ > 0 and ep ep(cδ ) for some absolute constant C. Then any A {0,,..., } of size A = δ necessarily contains a (non-trivial) arithmetic
More informationChapter 1. Number of special form. 1.1 Introduction(Marin Mersenne) 1.2 The perfect number. See the book.
Chapter 1 Number of special form 1.1 Introduction(Marin Mersenne) See the book. 1.2 The perfect number Definition 1.2.1. A positive integer n is said to be perfect if n is equal to the sum of all its positive
More informationDiophantine triples in a Lucas-Lehmer sequence
Annales Mathematicae et Informaticae 49 (01) pp. 5 100 doi: 10.33039/ami.01.0.001 http://ami.uni-eszterhazy.hu Diophantine triples in a Lucas-Lehmer sequence Krisztián Gueth Lorand Eötvös University Savaria
More informationQUARTERNIONS AND THE FOUR SQUARE THEOREM
QUARTERNIONS AND THE FOUR SQUARE THEOREM JIA HONG RAY NG Abstract. The Four Square Theorem was proved by Lagrange in 1770: every positive integer is the sum of at most four squares of positive integers,
More informationHeron triangles which cannot be decomposed into two integer right triangles
Heron triangles which cannot be decomposed into two integer right triangles Paul Yiu Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431 yiu@fau.edu 41st Meeting
More informationHomework 6 Solution. Math 113 Summer 2016.
Homework 6 Solution. Math 113 Summer 2016. 1. For each of the following ideals, say whether they are prime, maximal (hence also prime), or neither (a) (x 4 + 2x 2 + 1) C[x] (b) (x 5 + 24x 3 54x 2 + 6x
More information1. Factorization Divisibility in Z.
8 J. E. CREMONA 1.1. Divisibility in Z. 1. Factorization Definition 1.1.1. Let a, b Z. Then we say that a divides b and write a b if b = ac for some c Z: a b c Z : b = ac. Alternatively, we may say that
More informationSolutions to Problem Set 4 - Fall 2008 Due Tuesday, Oct. 7 at 1:00
Solutions to 8.78 Problem Set 4 - Fall 008 Due Tuesday, Oct. 7 at :00. (a Prove that for any arithmetic functions f, f(d = f ( n d. To show the relation, we only have to show this equality of sets: {d
More informationIntroduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
More informationElliptic Curves and Mordell s Theorem
Elliptic Curves and Mordell s Theorem Aurash Vatan, Andrew Yao MIT PRIMES December 16, 2017 Diophantine Equations Definition (Diophantine Equations) Diophantine Equations are polynomials of two or more
More informationARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY OF THE DIOPHANTINE EQUATION 8x = y 2
International Conference in Number Theory and Applications 01 Department of Mathematics, Faculty of Science, Kasetsart University Speaker: G. K. Panda 1 ARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY
More informationElementary Number Theory Review. Franz Luef
Elementary Number Theory Review Principle of Induction Principle of Induction Suppose we have a sequence of mathematical statements P(1), P(2),... such that (a) P(1) is true. (b) If P(k) is true, then
More informationContents. 4 Arithmetic and Unique Factorization in Integral Domains. 4.1 Euclidean Domains and Principal Ideal Domains
Ring Theory (part 4): Arithmetic and Unique Factorization in Integral Domains (by Evan Dummit, 018, v. 1.00) Contents 4 Arithmetic and Unique Factorization in Integral Domains 1 4.1 Euclidean Domains and
More informationDefinition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively
6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise
More informationTHE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - FALL SESSION ADVANCED ALGEBRA I.
THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - FALL SESSION 2006 110.401 - ADVANCED ALGEBRA I. Examiner: Professor C. Consani Duration: take home final. No calculators allowed.
More informationSolution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = ,
Solution Sheet 2 1. (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = 3. 2. (i) gcd (97, 157) = 1 = 34 97 21 157, (ii) gcd (527, 697) = 17 = 4 527 3 697, (iii) gcd (2323, 1679) =
More informationARITHMETIC OF POSITIVE INTEGERS HAVING PRIME SUMS OF COMPLEMENTARY DIVISORS
Math. J. Okayama Univ. 60 (2018), 155 164 ARITHMETIC OF POSITIVE INTEGERS HAVING PRIME SUMS OF COMPLEMENTARY DIVISORS Kenichi Shimizu Abstract. We study a class of integers called SP numbers (Sum Prime
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 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 informationAlgebraic number theory Solutions to exercise sheet for chapter 4
Algebraic number theory Solutions to exercise sheet for chapter 4 Nicolas Mascot n.a.v.mascot@warwick.ac.uk), Aurel Page a.r.page@warwick.ac.uk) TAs: Chris Birkbeck c.d.birkbeck@warwick.ac.uk), George
More informationPMA225 Practice Exam questions and solutions Victor P. Snaith
PMA225 Practice Exam questions and solutions 2005 Victor P. Snaith November 9, 2005 The duration of the PMA225 exam will be 2 HOURS. The rubric for the PMA225 exam will be: Answer any four questions. You
More informationMATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.
MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same
More informationMath 547, Exam 1 Information.
Math 547, Exam 1 Information. 2/10/10, LC 303B, 10:10-11:00. Exam 1 will be based on: Sections 5.1, 5.2, 5.3, 9.1; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationx 3 2x = (x 2) (x 2 2x + 1) + (x 2) x 2 2x + 1 = (x 4) (x + 2) + 9 (x + 2) = ( 1 9 x ) (9) + 0
1. (a) i. State and prove Wilson's Theorem. ii. Show that, if p is a prime number congruent to 1 modulo 4, then there exists a solution to the congruence x 2 1 mod p. (b) i. Let p(x), q(x) be polynomials
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 informationLEGENDRE S THEOREM, LEGRANGE S DESCENT
LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +
More informationDiscrete Logarithms. Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set
Discrete Logarithms Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set Z/mZ = {[0], [1],..., [m 1]} = {0, 1,..., m 1} of residue classes modulo m is called
More informationHomework 7 solutions M328K by Mark Lindberg/Marie-Amelie Lawn
Homework 7 solutions M328K by Mark Lindberg/Marie-Amelie Lawn Problem 1: 4.4 # 2:x 3 + 8x 2 x 1 0 (mod 1331). a) x 3 + 8x 2 x 1 0 (mod 11). This does not break down, so trial and error gives: x = 0 : f(0)
More informationMath 319 Problem Set #2 Solution 14 February 2002
Math 39 Problem Set # Solution 4 February 00. (.3, problem 8) Let n be a positive integer, and let r be the integer obtained by removing the last digit from n and then subtracting two times the digit ust
More informationThe Fundamental Theorem of Arithmetic
Chapter 1 The Fundamental Theorem of Arithmetic 1.1 Primes Definition 1.1. We say that p N is prime if it has just two factors in N, 1 and p itself. Number theory might be described as the study of the
More information