16 The Quadratic Reciprocity Law

Size: px
Start display at page:

Download "16 The Quadratic Reciprocity Law"

Transcription

1 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 thing to know in general But Gauss noticed something remarkable, namely that knowing is euivalent to knowing ;they need not be eual however He found the recise law which governs this relationshi, called the Quadratic Recirocity Law Gauss was very roud of ths result and gave several roofs We will give one of his roofs, which incidentally introduces a very basic, ubiuitous sum in Mathematics called the Gauss sum We will also give an alternate roof, which is in some sendse more clever than the first, due to Eisenstein Theorem Gauss Quadratic recirocity Let, be distinct odd rimes Then Exlicitly, /// { 1, if or is 1 mod 4 1, if and are 3 mod 4 This theorem is very useful in comutations Examle It is not easy to comute mod 691 Better to use then: ;

2 Proof#1oftheorem:, odd rimes, Put ξ e πi C Then ξ 1, but ξ m 1ifm< ξ is called a rimitive th root of unity in C Allowersofξ will be on the unit circle In fact, we get a regular -gon by converting the oint 1,ξ,,ξ 1 cyclotong circledivision Put R {α a 0 + a 1 ξ + + a ξ 1 ξ 1 a 0,a 1,,a 1 Z} Clearly, R Z, hence R has 0 1 Let be in R Then 1 1 α a i ξ,β b i ξ i1 i1 1 α ± β a i + b i ξ i R i1 Since ξ 1,givenanyn Z we can write n l + r, 0 r 1by Euclidean algorithm in Z, and conclude that ξ n ξ r So R contains all the integral owers of ξ Then it also contains finite integral linear combinations of such owers Conseuently, αβ R if α, β R So R is very much like Z Itisa-dimensional analog of Z This allows us to define the divisibility in R To be recise, if α, β R, wesaythatβ divides α, β α iff γ R such that α βγ In articular, R, anditmakessensetoaskif divides some number in R Definition: Letα, β R Wesaythat α β mod iff a b inr

3 This allows us to do congruence arithmetic mod in R To study, Gauss introduced the following Gauss Sum : S a ξ a a mod Clearly, S R Aside Not art of roof of Quad Reci, but interesting S 1 a1 a ξ a + a ξ a 1 a ξ a So if and if ure read or im Lemma 1: Proof of Lemma 1: 1 1,S S 1 a1 a ξ a + ξ a R R 1 1, S R ir a mod a mod b mod S 1 1 a ξ a a ab c mod ξ c b ξ a+b a mod b mod ξ a ξ b b ξ b ac a 3

4 So S c mod c mod ξ c ξ c a mod a mod ac a a 1 a c, where a a 1mod But a 1 a c where fc? c 0mod: fc S 1 1 a mod f0 a 1 as a 0 mod c mod 1 a c a mod a 0 mod f0 1 c 0mod: Note that, in this case, the set ξ c fc, a 0mod 1 {1 a c a mod, a 0mod} 1 a c runs over elements of Z/ {1} exactly once Indeed, given any b Z/, b 1mod, we can solve a + b 1mod, and the solution is uniue Therefore, fc b mod b 1 mod b 4

5 We roved earlier that so when c 0mod Conseuently S 1 1 Claim: c mod ξc 0 Proof of claim: ξ c c mod 1 ξ 0 Proof of claim: c mod b mod fc c 1 mod c mod b 0 1 1, ξ c ξ c 0 c mod c mod c mod c 0 mod ξ c ξ c+1 ξ c mod ξ c ξ c 0 as claimed ξ c 1+ξ + + ξ 1 1 ξ 1 ξ 0sinceξ 1 By claim, S This roves Lemma 1 Lemma 1: S 1 1 5

6 Lemma : S 1 mod This haens in R mod Proof of Lemma : S a mod a mod a ξ a ξ a + w, w R a a a because a ±1 and is odd In other words, S a ξ a mod a mod Since, is invertible mod, andthemaa a is a ermutation of Z/, alsoa 0mod iffa 0mod so the sum over a mod can be relaced with the sume over a mod Write b for a mod Then a b mod, where 1mod S b mod But b Since 1mod, So b So gives S b b ξ b mod * 1 1 b ξ b b b mod S 6 mod

7 This is justified because S 1 S 0mod, mod which follows from lemma 1 ProofofTheorem: Comute S 1 in different ways On the one hand, by lemma 1, ie, 1 S 1 S Euler mod 1 1 S 1 mod, S On the other hand, by lemma, S 1 So, utting them together we get mod mod Last time, gave a roof of Quadratic Recirocity law More recisely we roved: Theorem Gauss Let, be distinct, odd rimes Then

8 Examle: Check if 9 is a suare mod 43: 9 and 43 are distinct odd rimes, so by definition 9 mod 43 iff byqrl, as9 5mod QRL So 9 mod 43 Remark: QRL tells you a way to know whether is a suare mod or not But when it is a suare, it gives no rocedure to find the suare root One can use QRL to check whether a number is a rime, similar to the way one uses Fermat s little theorem For examle, one can show that m 179 is not a rime by looking at y def mod 179 Note: So,ifmis a rime, y 11 modm 179 Since 179 1mod4,byQRL, as 11 3 mod 8 on the other hand, one can check using PARI, or by successively suaring mod m 179, that modm 8

9 This is art of a homework roblem Get a contradiction! So the only ossibility is that 179 is not a rime which is easy to verify directly as But this method is helful, when it works, for larger numbers A histoical remark: GHHardy went to see Ramanujan, when the latter was dying of TB in England Then Ramanujan asked Hardy if the number of the taxicab Hardy came in was an interesting number Hardy said No, not interesting, just 179 Ramanujan relied immediately, saying, On the contrary, the number is interesting because it is the first number which can be written as a sum of cubes in two different ways Indeed we have A second roof of uadratic reci Eisenstein Eisenstein s trignometric lemma Lemma: Letn be a ositive, odd integer Then sin nx sin x n 1 4 n 1 j1 sin x sin πj n Proof: U to us Hint: treat as a olynomial in sin x: Examle: n 3 sin 3x sinx + x LHS sin x sin x sin x cos x +cosxsin x sin x sinxcos x +1 sin xsinx sin x 1 sin x+1 sin x3 4sin x RHS 4sin x sin π }{{ 3 } 3/ 3 4sin x 4 sin x 3 4 9

10 Sketch of roof of lemma: Use induction on n to show that sin nx sin x f nsin x, where f n is a olynomial in sin x of degree n 1 f 0 t 1,f 3 t 3 4t, On the other hand, the RHS of lemma is also of the form g n sin x, where g n is the exlicitly given olynomial in sin x of degree n 1 So it suffices to show that f n and g n have the same roots and that the leading coefficient of f n is 4 n 1 So when we use induction on n, check that the leading coefficient is 4 n 1 and that its roots are { sin πj n 1 j n 1 } Alternatively, check the constant coefficient by checking at x 0 Recall Gauss lemma: e s s S where S {1,,, 1 } and e x {±1} defined by s e s s, with s S Alying sin π, we get πs πes s sin sin πs e s sin since sin is an odd function So e s sin sin πs πs 10

11 By Gauss lemma, sin πs πs s S sin s S sin πs s S sin πs Note the ma S S is a ermutation of S So, πs sin πs sin s S s S 1 i1 sin πi sin πi 1 Alying Eisenstein s trig lemma with n and sub in 3, we get i 1 1 i 1 πi πj sin sin Can get everything we need from this without comuting the sines: Reversing the roles of and, weget i 1 i 1 Comaring 3 and 4, we see that πj πi sin sin

Math 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,

Math 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 information

MATH 3240Q Introduction to Number Theory Homework 7

MATH 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 information

RECIPROCITY LAWS JEREMY BOOHER

RECIPROCITY 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 information

MATH342 Practice Exam

MATH342 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 information

Quadratic Reciprocity

Quadratic 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 information

MATH 371 Class notes/outline October 15, 2013

MATH 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 information

x 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,

x 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 information

Primes - Problem Sheet 5 - Solutions

Primes - Problem Sheet 5 - Solutions Primes - Problem Sheet 5 - Solutions Class number, and reduction of quadratic forms Positive-definite Q1) Aly the roof of Theorem 5.5 to find reduced forms equivalent to the following, also give matrices

More information

MA3H1 TOPICS IN NUMBER THEORY PART III

MA3H1 TOPICS IN NUMBER THEORY PART III MA3H1 TOPICS IN NUMBER THEORY PART III SAMIR SIKSEK 1. Congruences Modulo m In quadratic recirocity we studied congruences of the form x 2 a (mod ). We now turn our attention to situations where is relaced

More information

QUADRATIC RECIPROCITY

QUADRATIC 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 information

t s (p). An Introduction

t 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 information

Practice Final Solutions

Practice 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 information

HENSEL S LEMMA KEITH CONRAD

HENSEL 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 information

MATH 361: NUMBER THEORY ELEVENTH LECTURE

MATH 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 information

QUADRATIC RECIPROCITY

QUADRATIC 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 information

Math 104B: Number Theory II (Winter 2012)

Math 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 information

MATH 361: NUMBER THEORY EIGHTH LECTURE

MATH 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 information

Math 261 Exam 2. November 7, The use of notes and books is NOT allowed.

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 information

Introduction to Arithmetic Geometry Fall 2013 Lecture #10 10/8/2013

Introduction 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 information

MATH 2710: NOTES FOR ANALYSIS

MATH 2710: NOTES FOR ANALYSIS MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite

More information

Jacobi symbols and application to primality

Jacobi symbols and application to primality Jacobi symbols and alication to rimality Setember 19, 018 1 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

More information

Quadratic 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

Quadratic 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 information

Number Theory Naoki Sato

Number Theory Naoki Sato Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO

More information

Practice Final Solutions

Practice 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 information

HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES

HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES HASSE INVARIANTS FOR THE CLAUSEN ELLIPTIC CURVES AHMAD EL-GUINDY AND KEN ONO Astract. Gauss s F x hyergeometric function gives eriods of ellitic curves in Legendre normal form. Certain truncations of this

More information

MA3H1 Topics in Number Theory. Samir Siksek

MA3H1 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 information

6 Binary Quadratic forms

6 Binary Quadratic forms 6 Binary Quadratic forms 6.1 Fermat-Euler Theorem A binary quadratic form is an exression of the form f(x,y) = ax 2 +bxy +cy 2 where a,b,c Z. Reresentation of an integer by a binary quadratic form has

More information

(Workshop on Harmonic Analysis on symmetric spaces I.S.I. Bangalore : 9th July 2004) B.Sury

(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 information

CERIAS Tech Report The period of the Bell numbers modulo a prime by Peter Montgomery, Sangil Nahm, Samuel Wagstaff Jr Center for Education

CERIAS 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 information

The Euler Phi Function

The Euler Phi Function The Euler Phi Function 7-3-2006 An arithmetic function takes ositive integers as inuts and roduces real or comlex numbers as oututs. If f is an arithmetic function, the divisor sum Dfn) is the sum of the

More information

MAT 311 Solutions to Final Exam Practice

MAT 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 information

The 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 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 information

We collect some results that might be covered in a first course in algebraic number theory.

We 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 information

Excerpt from "Intermediate Algebra" 2014 AoPS Inc.

Excerpt 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 information

HOMEWORK # 4 MARIA SIMBIRSKY SANDY ROGERS MATTHEW WELSH

HOMEWORK # 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 information

ON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS

ON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS #A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,

More information

p-adic Properties of Lengyel s Numbers

p-adic Properties of Lengyel s Numbers 1 3 47 6 3 11 Journal of Integer Sequences, Vol. 17 (014), Article 14.7.3 -adic Proerties of Lengyel s Numbers D. Barsky 7 rue La Condamine 75017 Paris France barsky.daniel@orange.fr J.-P. Bézivin 1, Allée

More information

A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract

A 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 information

Classification of Finite Fields

Classification of Finite Fields Classification of Finite Fields In these notes we use the properties of the polynomial x pd x to classify finite fields. The importance of this polynomial is explained by the following basic proposition.

More information

Mobius Functions, Legendre Symbols, and Discriminants

Mobius 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 information

ERRATA 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 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 information

2 Asymptotic density and Dirichlet density

2 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 information

2 Asymptotic density and Dirichlet density

2 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 information

RINGS OF INTEGERS WITHOUT A POWER BASIS

RINGS 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 information

arxiv: v1 [math.nt] 9 Sep 2015

arxiv: 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 information

Pythagorean triples and sums of squares

Pythagorean triples and sums of squares Pythagorean triles and sums of squares Robin Chaman 16 January 2004 1 Pythagorean triles A Pythagorean trile (x, y, z) is a trile of ositive integers satisfying z 2 + y 2 = z 2. If g = gcd(x, y, z) then

More information

ON THE SET a x + b g x (mod p) 1 Introduction

ON THE SET a x + b g x (mod p) 1 Introduction PORTUGALIAE MATHEMATICA Vol 59 Fasc 00 Nova Série ON THE SET a x + b g x (mod ) Cristian Cobeli, Marian Vâjâitu and Alexandru Zaharescu Abstract: Given nonzero integers a, b we rove an asymtotic result

More information

Frobenius Elements, the Chebotarev Density Theorem, and Reciprocity

Frobenius 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 information

Algebraic Number Theory

Algebraic 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 information

Representing Integers as the Sum of Two Squares in the Ring Z n

Representing Integers as the Sum of Two Squares in the Ring Z n 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.4 Reresenting Integers as the Sum of Two Squares in the Ring Z n Joshua Harrington, Lenny Jones, and Alicia Lamarche Deartment

More information

f(r) = a d n) d + + a0 = 0

f(r) = a d n) d + + a0 = 0 Math 400-00/Foundations of Algebra/Fall 07 Polynomials at the Foundations: Roots Next, we turn to the notion of a root of a olynomial in Q[x]. Definition 8.. r Q is a rational root of fx) Q[x] if fr) 0.

More information

PROBLEM 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. 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 information

Complex Analysis Homework 1

Complex Analysis Homework 1 Comlex Analysis Homework 1 Steve Clanton Sarah Crimi January 27, 2009 Problem Claim. If two integers can be exressed as the sum of two squares, then so can their roduct. Proof. Call the two squares that

More information

On generalizing happy numbers to fractional base number systems

On generalizing happy numbers to fractional base number systems On generalizing hay numbers to fractional base number systems Enriue Treviño, Mikita Zhylinski October 17, 018 Abstract Let n be a ositive integer and S (n) be the sum of the suares of its digits. It is

More information

Algebraic number theory LTCC Solutions to Problem Sheet 2

Algebraic 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 information

1 Integers and the Euclidean algorithm

1 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 information

Chapter 3. Number Theory. Part of G12ALN. Contents

Chapter 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 information

#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS

#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS #A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt

More information

A generalization of Scholz s reciprocity law

A generalization of Scholz s reciprocity law Journal de Théorie des Nombres de Bordeaux 19 007, 583 59 A generalization of Scholz s recirocity law ar Mark BUDDEN, Jeremiah EISENMENGER et Jonathan KISH Résumé. Nous donnons une généralisation de la

More information

The Jacobi Symbol. q q 1 q 2 q n

The 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 information

Sets of Real Numbers

Sets of Real Numbers Chater 4 Sets of Real Numbers 4. The Integers Z and their Proerties In our revious discussions about sets and functions the set of integers Z served as a key examle. Its ubiquitousness comes from the fact

More information

RESEARCH STATEMENT THOMAS WRIGHT

RESEARCH STATEMENT THOMAS WRIGHT RESEARCH STATEMENT THOMAS WRIGHT My research interests lie in the field of number theory, articularly in Diohantine equations, rime gas, and ellitic curves. In my thesis, I examined adelic methods for

More information

.4. Congruences. We say that a is congruent to b modulo N i.e. a b mod N i N divides a b or equivalently i a%n = b%n. So a is congruent modulo N to an

.4. Congruences. We say that a is congruent to b modulo N i.e. a b mod N i N divides a b or equivalently i a%n = b%n. So a is congruent modulo N to an . Modular arithmetic.. Divisibility. Given ositive numbers a; b, if a 6= 0 we can write b = aq + r for aroriate integers q; r such that 0 r a. The number r is the remainder. We say that a divides b (or

More information

A Curious Property of the Decimal Expansion of Reciprocals of Primes

A Curious Property of the Decimal Expansion of Reciprocals of Primes A Curious Proerty of the Decimal Exansion of Recirocals of Primes Amitabha Triathi January 6, 205 Abstract For rime 2, 5, the decimal exansion of / is urely eriodic. For those rime for which the length

More information

By Evan Chen OTIS, Internal Use

By 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 information

Characteristics of Fibonacci-type Sequences

Characteristics of Fibonacci-type Sequences Characteristics of Fibonacci-tye Sequences Yarden Blausa May 018 Abstract This aer resents an exloration of the Fibonacci sequence, as well as multi-nacci sequences and the Lucas sequence. We comare and

More information

QUADRATIC RECIPROCITY

QUADRATIC 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 information

On the Multiplicative Order of a n Modulo n

On 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 information

Jonathan Sondow 209 West 97th Street, New York, New York

Jonathan Sondow 209 West 97th Street, New York, New York #A34 INTEGERS 11 (2011) REDUCING THE ERDŐS-MOSER EQUATION 1 n + 2 n + + k n = (k + 1) n MODULO k AND k 2 Jonathan Sondow 209 West 97th Street, New York, New York jsondow@alumni.rinceton.edu Kieren MacMillan

More information

π(x) π( x) = x<n x gcd(n,p)=1 The sum can be extended to all n x, except that now the number 1 is included in the sum, so π(x) π( x)+1 = n x

π(x) π( x) = x<n x gcd(n,p)=1 The sum can be extended to all n x, except that now the number 1 is included in the sum, so π(x) π( x)+1 = n x Math 05 notes, week 7 C. Pomerance Sieving An imortant tool in elementary/analytic number theory is sieving. Let s begin with something familiar: the sieve of Ertatosthenes. This is usually introduced

More information

Small Zeros of Quadratic Forms Mod P m

Small 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 information

Elementary Analysis in Q p

Elementary Analysis in Q p Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some

More information

RATIONAL RECIPROCITY LAWS

RATIONAL RECIPROCITY LAWS RATIONAL RECIPROCITY LAWS MARK BUDDEN 1 10/7/05 The urose of this note is to rovide an overview of Rational Recirocity and in articular, of Scholz s recirocity law for the non-number theorist. In the first

More information

Series Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form.

Series Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form. Series Handout A. Determine which of the following sums are geometric. If the sum is geometric, exress the sum in closed form. 70 a) k= ( k ) b) 50 k= ( k )2 c) 60 k= ( k )k d) 60 k= (.0)k/3 2. Find the

More information

CS 6260 Some number theory. Groups

CS 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 information

Analytic number theory and quadratic reciprocity

Analytic 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 information

Quadratic Reciprocity. As in the previous notes, we consider the Legendre Symbol, defined by

Quadratic Reciprocity. As in the previous notes, we consider the Legendre Symbol, defined by Math 0 Sring 01 Quadratic Recirocity As in the revious notes we consider the Legendre Sybol defined by $ ˆa & 0 if a 1 if a is a quadratic residue odulo. % 1 if a is a quadratic non residue We also had

More information

DISCRIMINANTS IN TOWERS

DISCRIMINANTS 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 information

Factor Rings and their decompositions in the Eisenstein integers Ring Z [ω]

Factor 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 information

Verifying Two Conjectures on Generalized Elite Primes

Verifying Two Conjectures on Generalized Elite Primes 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.7 Verifying Two Conjectures on Generalized Elite Primes Xiaoqin Li 1 Mathematics Deartment Anhui Normal University Wuhu 241000,

More information

On the Rank of the Elliptic Curve y 2 = x(x p)(x 2)

On the Rank of the Elliptic Curve y 2 = x(x p)(x 2) On the Rank of the Ellitic Curve y = x(x )(x ) Jeffrey Hatley Aril 9, 009 Abstract An ellitic curve E defined over Q is an algebraic variety which forms a finitely generated abelian grou, and the structure

More information

DIRICHLET S THEOREM ON PRIMES IN ARITHMETIC PROGRESSIONS. 1. Introduction

DIRICHLET 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 information

MATH 371 Class notes/outline September 24, 2013

MATH 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 information

#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS

#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS #A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,

More information

Topic 7: Using identity types

Topic 7: Using identity types Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules

More information

THE LEAST PRIME QUADRATIC NONRESIDUE IN A PRESCRIBED RESIDUE CLASS MOD 4

THE LEAST PRIME QUADRATIC NONRESIDUE IN A PRESCRIBED RESIDUE CLASS MOD 4 THE LEAST PRIME QUADRATIC NONRESIDUE IN A PRESCRIBED RESIDUE CLASS MOD 4 PAUL POLLACK Abstract For all rimes 5, there is a rime quadratic nonresidue q < with q 3 (mod 4 For all rimes 3, there is a rime

More information

PRIME NUMBERS YANKI LEKILI

PRIME NUMBERS YANKI LEKILI PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers:,2,..., These are constructed using Peano axioms. We will not get into the hilosohical questions related to this and simly assume the

More information

Catalan s Equation Has No New Solution with Either Exponent Less Than 10651

Catalan s Equation Has No New Solution with Either Exponent Less Than 10651 Catalan s Euation Has No New Solution with Either Exonent Less Than 065 Maurice Mignotte and Yves Roy CONTENTS. Introduction and Overview. Bounding One Exonent as a Function of the Other 3. An Alication

More information

MAS 4203 Number Theory. M. Yotov

MAS 4203 Number Theory. M. Yotov MAS 4203 Number Theory M. Yotov June 15, 2017 These Notes were comiled by the author with the intent to be used by his students as a main text for the course MAS 4203 Number Theory taught at the Deartment

More information

Maths 4 Number Theory Notes 2012 Chris Smyth, University of Edinburgh ed.ac.uk

Maths 4 Number Theory Notes 2012 Chris Smyth, University of Edinburgh ed.ac.uk Maths 4 Number Theory Notes 202 Chris Smyth, University of Edinburgh c.smyth @ ed.ac.uk 0. Reference books There are no books I know of that contain all the material of the course. however, there are many

More information

Fermat's Little Theorem

Fermat'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 information

Congruences modulo 3 for two interesting partitions arising from two theta function identities

Congruences modulo 3 for two interesting partitions arising from two theta function identities Note di Matematica ISSN 113-53, e-issn 1590-093 Note Mat. 3 01 no., 1 7. doi:10.185/i1590093v3n1 Congruences modulo 3 for two interesting artitions arising from two theta function identities Kuwali Das

More information

19th Bay Area Mathematical Olympiad. Problems and Solutions. February 28, 2017

19th Bay Area Mathematical Olympiad. Problems and Solutions. February 28, 2017 th Bay Area Mathematical Olymiad February, 07 Problems and Solutions BAMO- and BAMO- are each 5-question essay-roof exams, for middle- and high-school students, resectively. The roblems in each exam are

More information

Elementary Number Theory

Elementary 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 information

Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively

Definition 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 information

Factorability in the ring Z[ 5]

Factorability in the ring Z[ 5] University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Paers in Mathematics Mathematics, Deartment of 4-2004 Factorability in the ring

More information

GAUSSIAN INTEGERS HUNG HO

GAUSSIAN 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 information

Primes of the form ±a 2 ± qb 2

Primes of the form ±a 2 ± qb 2 Stud. Univ. Babeş-Bolyai Math. 58(2013), No. 4, 421 430 Primes of the form ±a 2 ± qb 2 Eugen J. Ionascu and Jeff Patterson To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. Reresentations

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two oerations defined on them, addition and multilication,

More information

MATH 152 NOTES MOOR XU NOTES FROM A COURSE BY KANNAN SOUNDARARAJAN

MATH 152 NOTES MOOR XU NOTES FROM A COURSE BY KANNAN SOUNDARARAJAN MATH 5 NOTES MOOR XU NOTES FROM A COURSE BY KANNAN SOUNDARARAJAN Abstract These notes were taken from math 5 (Elementary Theory of Numbers taught by Kannan Soundararajan in Fall 00 at Stanford University

More information