Jacobi symbols and application to primality
|
|
- Dinah Morris
- 5 years ago
- Views:
Transcription
1 Jacobi symbols and alication to rimality Setember 19, The grou Z/Z We review the structure of the abelian grou Z/Z. Using Chinese remainder theorem, we can restrict to the case when = k is a rime ower. If k = 1 the grou is cyclic. Assume k. The cardinality of Z/ k Z is k. Since and k are corime, the grou Z/ k Z is the direct roduct of two subgrous with resective orders 1 and k. One can be more recise. We have the eact seuence 1 U 1 Z/ k Z F 1 1 where U 1 is the subgrou of all mod k such that 1 mod. Let V be the grou of solutions to the euation = 1. According to Hensel lemma, there are at least 1 such roots, and reduction modulo is a bijection from V onto F. The intersection of V and U 1 is trivial. For every n 1 let U n Z/Z be the subgrou consisting of all residues congruent to 1 modulo n. So {1} = U k U k... U 1. For every 1 n k 1, the uotient U n /U n+1 is cyclic of order and 1 + n is a generator of it. Indeed, the ma 1 + a n mod n+1 a mod is and isomorhism from U n /U n+1, onto Z/Z, +. Lemma 1 Let n be an integer such that 1 n k if 3 and n k if =. Let U n U n+1. Then U n+1 U n+. Indeed = 1 + a n and a is rime to. If 3 one comutes = 1+a n = 1+a n+1 + a m nm +a n 1+a n+1 mod n+ m since n n +. If = and n then m = 1 + a n = 1 + a n+1 + a n 1 + a n+1 mod n+ 1
2 since n n +. We deduce that if 3 then U 1 is cyclic of order k and 1 + is a generator. For =, we only rove that U is cyclic of order k and 5 is a generator. If is odd the grou Z/ k Z is isomorhic to Z/ 1Z Z/ k Z. For = one checks that U 1 = {1, } U so Z/ k Z is isomorhic to Z/Z Z/ k Z. The Legendre symbol Let be and odd rime. For every integer one defines the Legendre symbol as follows : 1. = 0 if divides,. = 1 if is a non-zero suare modulo, 3. = if is not a suare modulo. The ma is a grou homomorhism from F onto {1, }. One checks that = mod. So we obtain a first method to comute this Legendre symbol. The famous uadratic recirocity law states that Theorem 1 If and are two odd ositive distinct rimes then = 4. There are many roofs for this theorem. For eamle set Φ = and let A F [] be an irreducible factor of Φ modulo. Set L = F []/A and let ζ = mod A L. This is a -th root of unity in the field L. Question 1 Show that ζ is a rimitive -th root of unity its multilicative order is eactly.
3 The so called Gauss sum τ = F ζ is an element of the field L. One can show that τ = L. So τ is a suare root of in the algebraic closure of F. This suare root is in F if and only if τ = τ. On checks that τ = τ. So is a suare modulo if and only if = 1. This finishes the roof. We shall need also the following theorem Theorem For an odd rime = 8. Observe that if is an odd integer then = 1 + k and k + 1 = 1 + 4kk + 1 = is congruent to 1 modulo 8. And kk + 1/ is even if and only if k is congruent to 0 or 3 modulo 4 that is congruent to 1 or 7 modulo 8. ow let A F [] be an irreducible factor of modulo and set ζ = mod A the class of in F []/A. Question Prove that ζ is a rimitive 8-th root of 1. One checks that ζ + ζ =. So we have a suare root of in the algebraic closure of F. So is a suare if and only if this suare root is in F that is α = α. But α = ζ + ζ where the eonents only matter modulo 8. If is congruent to 1 or modulo 8 one deduces that α = α. If is congruent to 3 or 5 modulo 8 one checks that α = α. This roves formula and the theorem. 3 The Jacobi symbol Assume 3 is an odd integer and let = i e i i its rime decomosition. The Jacobi symbol is defined as a generalization of the Legendre symbol. One sets = ei. i i 3
4 This symbol only deends on the congruence class of modulo. It has many evident multilicative roerties inherited from the Lengendre symbol. For a eamle = 0 if and only if a are b not corime. b The uadratic recirocity law etends to this symbol. Theorem 3 Gauss Let M 3 and 3 two odd corime integers. One has = M, = M 8, and M M M = M 4. M Thanks to this theorem we can uickly comute the Jacobi symbol by successive Euclidean divisions. ote that if is not a rime, the Jacobi symbol does not distinguish uadratic residues. For eamle if = is the roduct of two odd rimes and if is rime to then = 1 means that either is a suare modulo and modulo, or that is not a suare modulo nor modulo. In the latter case one sometimes says that is a false suare. 4 The Solovay-Strassen rimality test Let be an odd integer. Let χ 1 : Z/Z Z/Z and χ : Z/Z Z/Z be the two grou homomorhisms defined by and χ 1 : mod χ : mod. We set χ 0 = χ /χ 1. It is evident that χ 0 is trivial if is a rime. One has the Lemma If is odd and comosite, then there eists an mod in Z/Z such that χ 0 1. Assume first that is divisible by a non-trivial suare : there eists an odd rime and an integer k such that k divides eactly. Set M = / k. Let G Z/Z be the subgrou consisting of all residues congruent to 1 modulo M. This is a cyclic grou of order k. The restriction of the Jacobi symbol to this sub-grou is trivial. The restriction of χ 1 is not because is rime to. Assume now that is suare-free. Let be an odd rime factor of and set M = /. Let be an integer congruent to 1 modulo M and which is not a suare modulo. Then χ = and χ 1 = 1 mod M. So χ 1 χ. 4
5 If is an odd comosite integer then the kernel of χ 0 is a strict subgrou of Z/Z. Its cardinality is. We have at least one chance over two to find χ 0 1 if is chosen at random uniformly in Z/Z. Since we have olynomial time algorithms to comute χ 1 and χ we obtain a robabilistic rimality test : 1. check that is odd;. ick at random in Z/Z and comute χ 1 and χ ; 3. if χ 1 χ, one knows that is comosite; 4. if χ 1 = χ, one cannot conclude... but one can try again! If is odd and comosite and if Z/Z is such that χ 1 = χ, one says that is a false witness. The roortion of false witnesses is at most 1/. 5
Math 261 Exam 2. November 7, The use of notes and books is NOT allowed.
Math 261 Eam 2 ovember 7, 2018 The use of notes and books is OT allowed Eercise 1: Polynomials mod 691 (30 ts In this eercise, you may freely use the fact that 691 is rime Consider the olynomials f( 4
More informationSQUARES IN Z/NZ. q = ( 1) (p 1)(q 1)
SQUARES I Z/Z We study squares in the ring Z/Z from a theoretical and comutational oint of view. We resent two related crytograhic schemes. 1. SQUARES I Z/Z Consider for eamle the rime = 13. Write the
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 informationRECIPROCITY LAWS JEREMY BOOHER
RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationAlgebraic number theory LTCC Solutions to Problem Sheet 2
Algebraic number theory LTCC 008 Solutions to Problem Sheet ) Let m be a square-free integer and K = Q m). The embeddings K C are given by σ a + b m) = a + b m and σ a + b m) = a b m. If m mod 4) then
More informationMATH 371 Class notes/outline October 15, 2013
MATH 371 Class notes/outline October 15, 2013 More on olynomials We now consider olynomials with coefficients in rings (not just fields) other than R and C. (Our rings continue to be commutative and have
More informationWe collect some results that might be covered in a first course in algebraic number theory.
1 Aendices We collect some results that might be covered in a first course in algebraic number theory. A. uadratic Recirocity Via Gauss Sums A1. Introduction In this aendix, is an odd rime unless otherwise
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013 In this lecture we lay the groundwork needed to rove the Hasse-Minkowski theorem for Q, which states that a quadratic form over
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCIT POOJA PATEL Abstract. This aer is an self-contained exosition of the law of uadratic recirocity. We will give two roofs of the Chinese remainder theorem and a roof of uadratic recirocity.
More informationMATH 3240Q Introduction to Number Theory Homework 7
As long as algebra and geometry have been searated, their rogress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched
More informationClass Field Theory. Peter Stevenhagen. 1. Class Field Theory for Q
Class Field Theory Peter Stevenhagen Class field theory is the study of extensions Q K L K ab K = Q, where L/K is a finite abelian extension with Galois grou G. 1. Class Field Theory for Q First we discuss
More informationt s (p). An Introduction
Notes 6. Quadratic Gauss Sums Definition. Let a, b Z. Then we denote a b if a divides b. Definition. Let a and b be elements of Z. Then c Z s.t. a, b c, where c gcda, b max{x Z x a and x b }. 5, Chater1
More informationFrobenius Elements, the Chebotarev Density Theorem, and Reciprocity
Frobenius Elements, the Chebotarev Density Theorem, and Recirocity Dylan Yott July 30, 204 Motivation Recall Dirichlet s theorem from elementary number theory. Theorem.. For a, m) =, there are infinitely
More informationx 2 a mod m. has a solution. Theorem 13.2 (Euler s Criterion). Let p be an odd prime. The congruence x 2 1 mod p,
13. Quadratic Residues We now turn to the question of when a quadratic equation has a solution modulo m. The general quadratic equation looks like ax + bx + c 0 mod m. Assuming that m is odd or that b
More informationLECTURE 10: JACOBI SYMBOL
LECTURE 0: JACOBI SYMBOL The Jcobi symbol We wish to generlise the Legendre symbol to ccomodte comosite moduli Definition Let be n odd ositive integer, nd suose tht s, where the i re rime numbers not necessrily
More informationQuadratic Reciprocity
Quadratic Recirocity 5-7-011 Quadratic recirocity relates solutions to x = (mod to solutions to x = (mod, where and are distinct odd rimes. The euations are oth solvale or oth unsolvale if either or has
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationMATH 361: NUMBER THEORY EIGHTH LECTURE
MATH 361: NUMBER THEORY EIGHTH LECTURE 1. Quadratic Recirocity: Introduction Quadratic recirocity is the first result of modern number theory. Lagrange conjectured it in the late 1700 s, but it was first
More informationQUADRATIC RECIPROCITY
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this roject are to have the reader(s) gain an areciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein s slick
More informationCS 6260 Some number theory. Groups
Let Z = {..., 2, 1, 0, 1, 2,...} denote the set of integers. Let Z+ = {1, 2,...} denote the set of ositive integers and = {0, 1, 2,...} the set of non-negative integers. If a, are integers with > 0 then
More informationarxiv: v2 [math.nt] 9 Oct 2018
ON AN EXTENSION OF ZOLOTAREV S LEMMA AND SOME PERMUTATIONS LI-YUAN WANG AND HAI-LIANG WU arxiv:1810.03006v [math.nt] 9 Oct 018 Abstract. Let be an odd rime, for each integer a with a, the famous Zolotarev
More informationThe Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001
The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001 The Hasse-Minkowski Theorem rovides a characterization of the rational quadratic forms. What follows is a roof of the Hasse-Minkowski
More informationMobius Functions, Legendre Symbols, and Discriminants
Mobius Functions, Legendre Symbols, and Discriminants 1 Introduction Zev Chonoles, Erick Knight, Tim Kunisky Over the integers, there are two key number-theoretic functions that take on values of 1, 1,
More informationFactor Rings and their decompositions in the Eisenstein integers Ring Z [ω]
Armenian Journal of Mathematics Volume 5, Number 1, 013, 58 68 Factor Rings and their decomositions in the Eisenstein integers Ring Z [ω] Manouchehr Misaghian Deartment of Mathematics, Prairie View A&M
More informationMATH 371 Class notes/outline September 24, 2013
MATH 371 Class notes/outline Setember 24, 2013 Rings Armed with what we have looked at for the integers and olynomials over a field, we re in a good osition to take u the general theory of rings. Definitions:
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationDISCRIMINANTS IN TOWERS
DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K/F be a finite searable extension field, and let B be the integral closure of A in K. In this note, we will
More informationMAT 311 Solutions to Final Exam Practice
MAT 311 Solutions to Final Exam Practice Remark. If you are comfortable with all of the following roblems, you will be very well reared for the midterm. Some of the roblems below are more difficult than
More informationMATH 361: NUMBER THEORY ELEVENTH LECTURE
MATH 361: NUMBER THEORY ELEVENTH LECTURE The subjects of this lecture are characters, Gauss sums, Jacobi sums, and counting formulas for olynomial equations over finite fields. 1. Definitions, Basic Proerties
More informationHOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH
HOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH 1. Section 2.1, Problems 5, 8, 28, and 48 Problem. 2.1.5 Write a single congruence that is equivalent to the air of congruences x 1 mod 4 and x 2
More information2 Asymptotic density and Dirichlet density
8.785: Analytic Number Theory, MIT, sring 2007 (K.S. Kedlaya) Primes in arithmetic rogressions In this unit, we first rove Dirichlet s theorem on rimes in arithmetic rogressions. We then rove the rime
More informationTHE THEORY OF NUMBERS IN DEDEKIND RINGS
THE THEORY OF NUMBERS IN DEDEKIND RINGS JOHN KOPPER Abstract. This aer exlores some foundational results of algebraic number theory. We focus on Dedekind rings and unique factorization of rime ideals,
More information3 Properties of Dedekind domains
18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following
More informationPartII Number Theory
PartII Number Theory zc3 This is based on the lecture notes given by Dr.T.A.Fisher, with some other toics in number theory (ossibly not covered in the lecture). Some of the theorems here are non-examinable.
More informationChapter 3. Number Theory. Part of G12ALN. Contents
Chater 3 Number Theory Part of G12ALN Contents 0 Review of basic concets and theorems The contents of this first section well zeroth section, really is mostly reetition of material from last year. Notations:
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 J. E. CREMONA Contents 0. Introduction: What is Number Theory? 2 Basic Notation 3 1. Factorization 4 1.1. Divisibility in Z 4 1.2. Greatest Common
More informationRINGS OF INTEGERS WITHOUT A POWER BASIS
RINGS OF INTEGERS WITHOUT A POWER BASIS KEITH CONRAD Let K be a number field, with degree n and ring of integers O K. When O K = Z[α] for some α O K, the set {1, α,..., α n 1 } is a Z-basis of O K. We
More informationQuadratic Residues, Quadratic Reciprocity. 2 4 So we may as well start with x 2 a mod p. p 1 1 mod p a 2 ±1 mod p
Lecture 9 Quadratic Residues, Quadratic Recirocity Quadratic Congruence - Consider congruence ax + bx + c 0 mod, with a 0 mod. This can be reduced to x + ax + b 0, if we assume that is odd ( is trivial
More informationHENSEL S LEMMA KEITH CONRAD
HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar
More informationLARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS CONTAINING SQUARE-FREE NUMBERS AND PERFECT POWERS OF PRIME NUMBERS
LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS CONTAINING SQUARE-FREE NUMBERS AND PERFECT POWERS OF PRIME NUMBERS HELMUT MAIER AND MICHAEL TH. RASSIAS Abstract. We rove a modification as well as an imrovement
More information(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury
Is e π 163 odd or even? (Worksho on Harmonic Analysis on symmetric saces I.S.I. Bangalore : 9th July 004) B.Sury e π 163 = 653741640768743.999999999999.... The object of this talk is to exlain this amazing
More informationElementary Number Theory
Elementary Number Theory WISB321 = F.Beukers 2012 Deartment of Mathematics UU ELEMENTARY NUMBER THEORY Frits Beukers Fall semester 2013 Contents 1 Integers and the Euclidean algorithm 4 1.1 Integers................................
More informationDIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS. 1. Introduction
DIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS INNA ZAKHAREVICH. Introduction It is a well-known fact that there are infinitely many rimes. However, it is less clear how the rimes are distributed
More informationPseudorandom Sequence Generation
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467a: Crytograhy and Comuter Security Handout #21 Professor M. J. Fischer November 29, 2005 Pseudorandom Seuence Generation 1 Distinguishability and
More informationAdvanced Cryptography Midterm Exam
Advanced Crytograhy Midterm Exam Solution Serge Vaudenay 17.4.2012 duration: 3h00 any document is allowed a ocket calculator is allowed communication devices are not allowed the exam invigilators will
More information1 Integers and the Euclidean algorithm
1 1 Integers and the Euclidean algorithm Exercise 1.1 Prove, n N : induction on n) 1 3 + 2 3 + + n 3 = (1 + 2 + + n) 2 (use Exercise 1.2 Prove, 2 n 1 is rime n is rime. (The converse is not true, as shown
More informationModeling Chebyshev s Bias in the Gaussian Primes as a Random Walk
Modeling Chebyshev s Bias in the Gaussian Primes as a Random Walk Daniel J. Hutama July 18, 2016 Abstract One asect of Chebyshev s bias is the henomenon that a rime number,, modulo another rime number,,
More informationMATH 210A, FALL 2017 HW 5 SOLUTIONS WRITTEN BY DAN DORE
MATH 20A, FALL 207 HW 5 SOLUTIONS WRITTEN BY DAN DORE (If you find any errors, lease email ddore@stanford.edu) Question. Let R = Z[t]/(t 2 ). Regard Z as an R-module by letting t act by the identity. Comute
More informationProbabilistic Algorithms
Probabilistic Algorithms Klaus Sutner Carnegie Mellon University Fall 2017 1 Some Probabilistic Algorithms Probabilistic Primality Testing RP and BPP Where Are We? 3 Examle 1: Order Statistics 4 We have
More informationThe Jacobi Symbol. q q 1 q 2 q n
The Jacobi Symbol It s a little inconvenient that the Legendre symbol a is only defined when the bottom is an odd p prime You can extend the definition to allow an odd positive number on the bottom using
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More informationANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM
ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to
More informationQUADRATIC FORMS, BASED ON (A COURSE IN ARITHMETIC BY SERRE)
QUADRATIC FORMS, BASED ON (A COURSE IN ARITHMETIC BY SERRE) HEE OH 1. Lecture 1:Introduction and Finite fields Let f be a olynomial with integer coefficients. One of the basic roblem is to understand if
More informationOn the Multiplicative Order of a n Modulo n
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 2010), Article 10.2.1 On the Multilicative Order of a n Modulo n Jonathan Chaelo Université Lille Nord de France F-59000 Lille France jonathan.chaelon@lma.univ-littoral.fr
More informationMath 104B: Number Theory II (Winter 2012)
Math 104B: Number Theory II (Winter 01) Alina Bucur Contents 1 Review 11 Prime numbers 1 Euclidean algorithm 13 Multilicative functions 14 Linear diohantine equations 3 15 Congruences 3 Primes as sums
More informationarxiv: v2 [math.nt] 11 Jun 2016
Congruent Ellitic Curves with Non-trivial Shafarevich-Tate Grous Zhangjie Wang Setember 18, 018 arxiv:1511.03810v [math.nt 11 Jun 016 Abstract We study a subclass of congruent ellitic curves E n : y x
More informationPractice Final Solutions
Practice Final Solutions 1. True or false: (a) If a is a sum of three squares, and b is a sum of three squares, then so is ab. False: Consider a 14, b 2. (b) No number of the form 4 m (8n + 7) can be written
More informationDegree in Mathematics
Degree in Mathematics Title: Gauss s roofs of the uadratic recirocity law Author: Anna Febrer Galvany Advisor: Jordi Quer Bosor Deartment: Matemàtica Alicada II Academic year: 01/013 Universitat Politècnica
More informationRECIPROCITY, BRAUER GROUPS AND QUADRATIC FORMS OVER NUMBER FIELDS
RECIPROCITY, BRAUER GROUPS AND QUADRATIC FORMS OVER NUMBER FIELDS THOMAS POGUNTKE Abstract. Artin s recirocity law is a vast generalization of quadratic recirocity and contains a lot of information about
More informationAlgebraic Number Theory
Algebraic Number Theory Joseh R. Mileti May 11, 2012 2 Contents 1 Introduction 5 1.1 Sums of Squares........................................... 5 1.2 Pythagorean Triles.........................................
More informationA CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod p) ZHI-HONG SUN
A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod ) ZHI-HONG SUN Deartment of Mathematics, Huaiyin Teachers College, Huaian 223001, Jiangsu, P. R. China e-mail: hyzhsun@ublic.hy.js.cn
More informationMersenne and Fermat Numbers
NUMBER THEORY CHARLES LEYTEM Mersenne and Fermat Numbers CONTENTS 1. The Little Fermat theorem 2 2. Mersenne numbers 2 3. Fermat numbers 4 4. An IMO roblem 5 1 2 CHARLES LEYTEM 1. THE LITTLE FERMAT THEOREM
More informationNumber Theory. Lectured by V. Neale Michaelmas Term 2011
Number Theory Lectured by V Neale Michaelmas Term 0 NUMBER THEORY C 4 lectures, Michaelmas term Page Page 5 Page Page 5 Page 9 Page 3 Page 4 Page 50 Page 54 Review from Part IA Numbers and Sets: Euclid
More informationPractice Final Solutions
Practice Final Solutions 1. Find integers x and y such that 13x + 1y 1 SOLUTION: By the Euclidean algorithm: One can work backwards to obtain 1 1 13 + 2 13 6 2 + 1 1 13 6 2 13 6 (1 1 13) 7 13 6 1 Hence
More informationExcerpt from "Intermediate Algebra" 2014 AoPS Inc.
Ecert from "Intermediate Algebra" 04 AoPS Inc. www.artofroblemsolving.com for which our grah is below the -ais with the oints where the grah intersects the -ais (because the ineuality is nonstrict), we
More informationMA3H1 Topics in Number Theory. Samir Siksek
MA3H1 Toics in Number Theory Samir Siksek Samir Siksek, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom E-mail address: samir.siksek@gmail.com Contents Chater 0. Prologue
More informationBy Evan Chen OTIS, Internal Use
Solutions Notes for DNY-NTCONSTRUCT Evan Chen January 17, 018 1 Solution Notes to TSTST 015/5 Let ϕ(n) denote the number of ositive integers less than n that are relatively rime to n. Prove that there
More informationGenus theory and the factorization of class equations over F p
arxiv:1409.0691v2 [math.nt] 10 Dec 2017 Genus theory and the factorization of class euations over F Patrick Morton March 30, 2015 As is well-known, the Hilbert class euation is the olynomial H D (X) whose
More informationMATH 242: Algebraic number theory
MATH 4: Algebraic number theory Matthew Morrow (mmorrow@math.uchicago.edu) Contents 1 A review of some algebra Quadratic residues and quadratic recirocity 4 3 Algebraic numbers and algebraic integers 1
More informationarxiv: v1 [math.nt] 9 Sep 2015
REPRESENTATION OF INTEGERS BY TERNARY QUADRATIC FORMS: A GEOMETRIC APPROACH GABRIEL DURHAM arxiv:5090590v [mathnt] 9 Se 05 Abstract In957NCAnkenyrovidedanewroofofthethreesuarestheorem using geometry of
More informationQUADRATIC RESIDUES AND DIFFERENCE SETS
QUADRATIC RESIDUES AND DIFFERENCE SETS VSEVOLOD F. LEV AND JACK SONN Abstract. It has been conjectured by Sárközy that with finitely many excetions, the set of quadratic residues modulo a rime cannot be
More informationAlmost All Palindromes Are Composite
Almost All Palindromes Are Comosite William D Banks Det of Mathematics, University of Missouri Columbia, MO 65211, USA bbanks@mathmissouriedu Derrick N Hart Det of Mathematics, University of Missouri Columbia,
More informationElliptic Curves Spring 2015 Problem Set #1 Due: 02/13/2015
18.783 Ellitic Curves Sring 2015 Problem Set #1 Due: 02/13/2015 Descrition These roblems are related to the material covered in Lectures 1-2. Some of them require the use of Sage, and you will need to
More informationGAUSSIAN INTEGERS HUNG HO
GAUSSIAN INTEGERS HUNG HO Abstract. We will investigate the ring of Gaussian integers Z[i] = {a + bi a, b Z}. First we will show that this ring shares an imortant roerty with the ring of integers: every
More informationWhen do Fibonacci invertible classes modulo M form a subgroup?
Calhoun: The NPS Institutional Archive DSace Reository Faculty and Researchers Faculty and Researchers Collection 2013 When do Fibonacci invertible classes modulo M form a subgrou? Luca, Florian Annales
More informationYALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467a: Crytograhy ad Comuter Security Notes 16 (rev. 1 Professor M. J. Fischer November 3, 2008 68 Legedre Symbol Lecture Notes 16 ( Let be a odd rime,
More informationERRATA AND SUPPLEMENTARY MATERIAL FOR A FRIENDLY INTRODUCTION TO NUMBER THEORY FOURTH EDITION
ERRATA AND SUPPLEMENTARY MATERIAL FOR A FRIENDLY INTRODUCTION TO NUMBER THEORY FOURTH EDITION JOSEPH H. SILVERMAN Acknowledgements Page vii Thanks to the following eole who have sent me comments and corrections
More informationCERIAS Tech Report The period of the Bell numbers modulo a prime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education
CERIAS Tech Reort 2010-01 The eriod of the Bell numbers modulo a rime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education and Research Information Assurance and Security Purdue University,
More information16 The Quadratic Reciprocity Law
16 The Quadratic Recirocity Law Fix an odd rime If is another odd rime, a fundamental uestion, as we saw in the revious section, is to know the sign, ie, whether or not is a suare mod This is a very hard
More informationAn Overview of Witt Vectors
An Overview of Witt Vectors Daniel Finkel December 7, 2007 Abstract This aer offers a brief overview of the basics of Witt vectors. As an alication, we summarize work of Bartolo and Falcone to rove that
More informationGalois representations on torsion points of elliptic curves NATO ASI 2014 Arithmetic of Hyperelliptic Curves and Cryptography
Galois reresentations on torsion oints of ellitic curves NATO ASI 04 Arithmetic of Hyerellitic Curves and Crytograhy Francesco Paalardi Ohrid, August 5 - Setember 5, 04 Lecture - Introduction Let /Q be
More informationIntroductory Number Theory
Introductory Number Theory Lecture Notes Sudita Mallik May, 208 Contents Introduction. Notation and Terminology.............................2 Prime Numbers.................................. 2 2 Divisibility,
More informationCONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION
CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION SILVIU RADU AND JAMES A. SELLERS Abstract. In this aer, we rove arithmetic roerties modulo 5 7 satisfied by the function odn which denotes the
More informationDiophantine Equations
Diohantine Equations Winter Semester 018/019 University of Bayreuth Michael Stoll Contents 1. Introduction and Examles 3. Aetizers 8 3. The Law of Quadratic Recirocity 1 Print version of October 5, 018,
More informationCDH/DDH-Based Encryption. K&L Sections , 11.4.
CDH/DDH-Based Encrytion K&L Sections 8.3.1-8.3.3, 11.4. 1 Cyclic grous A finite grou G of order q is cyclic if it has an element g of q. { 0 1 2 q 1} In this case, G = g = g, g, g,, g ; G is said to be
More informationGENERALIZED FACTORIZATION
GENERALIZED FACTORIZATION GRANT LARSEN Abstract. Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can imose on it to necessitate analogs of unique
More informationPROBLEM SET 5 SOLUTIONS. Solution. We prove that the given congruence equation has no solutions. Suppose for contradiction that. (x 2) 2 1 (mod 7).
PROBLEM SET 5 SOLUTIONS 1 Fid every iteger solutio to x 17x 5 0 mod 45 Solutio We rove that the give cogruece equatio has o solutios Suose for cotradictio that the equatio x 17x 5 0 mod 45 has a solutio
More informationSuper Congruences. Master s Thesis Mathematical Sciences
Suer Congruences Master s Thesis Mathematical Sciences Deartment of Mathematics Author: Thomas Attema Suervisor: Prof. Dr. Frits Beukers Second Reader: Prof. Dr. Gunther L.M. Cornelissen Abstract In 011
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationDIRICHLET S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS. Contents. 1. Dirichlet s theorem on arithmetic progressions
DIRICHLET S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS ANG LI Abstract. Dirichlet s theorem states that if q and l are two relatively rime ositive integers, there are infinitely many rimes of the
More informationAnalytic number theory and quadratic reciprocity
Analytic number theory and quadratic recirocity Levent Aloge March 31, 013 Abstract What could the myriad tools of analytic number theory for roving bounds on oscillating sums ossibly have to say about
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More informationClass number in non Galois quartic and non abelian Galois octic function fields over finite fields
Class number in non Galois quartic and non abelian Galois octic function fields over finite fields Yves Aubry G. R. I. M. Université du Sud Toulon-Var 83 957 La Garde Cedex France yaubry@univ-tln.fr Abstract
More informationMAT4250 fall 2018: Algebraic number theory (with a view toward arithmetic geometry)
MAT450 fall 018: Algebraic number theory (with a view toward arithmetic geometry) Håkon Kolderu Welcome to MAT450, a course on algebraic number theory. This fall we aim to cover the basic concets and results
More informationOn the Greatest Prime Divisor of N p
On the Greatest Prime Divisor of N Amir Akbary Abstract Let E be an ellitic curve defined over Q For any rime of good reduction, let E be the reduction of E mod Denote by N the cardinality of E F, where
More information#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS
#A37 INTEGERS 15 (2015) NOTE ON A RESULT OF CHUNG ON WEIL TYPE SUMS Norbert Hegyvári ELTE TTK, Eötvös University, Institute of Mathematics, Budaest, Hungary hegyvari@elte.hu François Hennecart Université
More informationUnit Groups of Semisimple Group Algebras of Abelian p-groups over a Field*
JOURNAL OF ALGEBRA 88, 580589 997 ARTICLE NO. JA96686 Unit Grous of Semisimle Grou Algebras of Abelian -Grous over a Field* Nako A. Nachev and Todor Zh. Mollov Deartment of Algebra, Plodi Uniersity, 4000
More informationSmall Zeros of Quadratic Forms Mod P m
International Mathematical Forum, Vol. 8, 2013, no. 8, 357-367 Small Zeros of Quadratic Forms Mod P m Ali H. Hakami Deartment of Mathematics, Faculty of Science, Jazan University P.O. Box 277, Jazan, Postal
More informationA FEW EQUIVALENCES OF WALL-SUN-SUN PRIME CONJECTURE
International Journal of Mathematics & Alications Vol 4, No 1, (June 2011), 77-86 A FEW EQUIVALENCES OF WALL-SUN-SUN PRIME CONJECTURE ARPAN SAHA AND KARTHIK C S ABSTRACT: In this aer, we rove a few lemmas
More information