MATH342 Practice Exam
|
|
- Beverley Jones
- 5 years ago
- Views:
Transcription
1 MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June It is not imlied that all questions on the real examination will follow the content of the ractice questions very closely. You should exect the unexected, esecially at the ends of exam questions, and think things out as best you can. 1. For x and y Z, and n Z + define x y mod n in terms of division by n. Prove that if gcd(n, m) = 1 then mx my mod n x y mod n. Find all integer solutions of the following. In some cases there may be no solutions. a) 3x 6 mod 9. b) 3x 5 mod 6. c) x 2 + x mod 5. d) x 2 1 mod 7. 3x 1 mod 7 e) Solve the simultaneous equations x 5 mod 6. 3x 2 mod 5 [20 marks] 1
2 2. If and q are ositive rimes with < q and n is comosite (that is, not rime) for all integers n with < n < q, then the set G(, q) = {n Z + : < n < q} is called a rime ga and its length is q. a) By considering the integers n! + k for 2 k n show that there is a rime ga of length n for each n Z +. Find the least ossible rime such that G(, q) is a rime ga of length 4. b) State the Fundamental Theorem of Arithmetic. Use it to show that, if n is comosite, then n is divisible by a rime with n. Use this to show that 89 and 97 are rime. c) Define the rime number counting function π(x). State the Prime Number Theorem. d) Let k (k 1) be an enumeration of all the ositive rimes with 1 = 2 and k < k+1 for all k 1 Prove that if n is sufficiently large, then there is a rime ga G( k, k+1 ) with k n and k+1 k > 1 ln n. 2 Hint: Let m be the largest integer with m n. Consider m k=1 ( k+1 k ) and aly the Prime Number Theorem to π(n). 3. State Fermat s Little Theorem. (i) Use it to rove that is divisible by 17. (ii) Find all the ossible orders of elements of the grou of units G 17. Find all rimitive elements. Give an examle of an element of each ossible order. (iii) Let m and n Z + with n 2, and let nm 1 be divisible by 17. Show n 1 that either m is even; or m 0 mod 17 and n 1 mod 17. Find all ossible values of n in the cases when m is even but not divisible by 4, or is divisible by 4 but not divisible by 8. 2
3 4. Define Euler s φ function. Prove that if is a ositive rime and a Z + then φ( a ) = a 1 ( 1). Also comute the sum a (using Euler s notation) of the divisors of a. Now write down the formulas for φ(n) and n, for any n Z +, with n 2, in terms of the rime decomosition of n, where n = m i=1 k i i for distinct rimes i and integers k i 1. Comute φ(2016) and φ(11!). Find all integers n such that φ(n) = Let n 1 2 and n 2 2 be corime integers. Show that, for integers x and y, x y mod n 1 n 2 (x y mod n 1 x y mod n 2 ). Give an examle to show that this is not true in general if n 1 and n 2 are not corime. Recall that n Z + is a seudorime to base a if a n 1 1 mod n, and n is a Carmichael number if n is comosite and is a seudorime to base a for all a in the grou G n of units mod n. Let n = = Verify that 2047 is a seudorime to base 2. Exlain why any rime dividing must be 1 mod 11 and find the rime factorisation of Give Korselt s equivalent definition of a Carmichael number. Use it to verify that 2821 is a Carmichael number. Now let n be any integer 2. Show that is a subgrou of G n. {a : a n 1 1 mod n} Now let n = 35. Identify all a mod 35 such that 35 is a seudorime to base a. 3
4 6. In this question let Z[i] be the ring of Gaussian integers, that is Z[i] = {s + it : s, t Z} a) Show that Z[i] is a Euclidean domain with Euclidean valuation v(s + it) = s 2 + t 2. Hint: Show that if z C then there is q Z[i] such that z q If a and b Z[i] with b 0, aly this with z = a/b. b) Show that if n, s, t Z and s + it divides n, then s it divides n and s 2 + t 2 divides n 2 in Z. Show also, using the fact that comlex conjugation is multilicative or otherwise, that if n 1 Z + and n 2 Z + are both the sums of the squares of two integers, then the same is true for n 1 n 2. c) Using the fact that Z[i] is a unique factorisation domain, show that if s and t are both non-zero integers and s + it is rime in Z[i], then s 2 + t 2 is rime in Z. Deduce that if s + it divides n in Z[i] then s 2 + t 2 divides n in Z. d) Exlain why any integer n which is a sum of two non-zero integer squares is the roduct of rimes with this roerty and a square of an integer. The square of the integer can be just 1 but there must be at least one rime in the roduct. e) Show that there are infinitely many rimes which are the sum of two squares. Hint: Suose there are just finitely many such rimes q i, 1 i n. Then let N 1 = n i=1 q i and N = N Consider any rime which divides N. 4
5 7. Define the Legendre symbol for any ositive rime and any integer q corime to. Show that if is any odd rime then q ( 1)/2 mod, stating any theory that you use. In articular, you may assume the existence of a rimitive element in G. Deduce that F : q mod : G {±1} is a grou homomorhism. ( ) 1 State the value of for any odd rime. State Gauss Law of quadratic recirocity for for any distinct ositive rimes q and, including the case q = 2. Deduce from these laws that for any odd rime ( ) { 2 1 if = 1 or 3 mod 8 = 1 if = 1 or 3 mod 8. Comute ( ) 5 19 and ( )
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 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 informationJacobi 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 informationMath 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 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 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 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 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 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 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 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 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 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 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 informationThe Arm Prime Factors Decomposition
The Arm Prime Factors Decomosition Arm Boris Nima arm.boris@gmail.com Abstract We introduce the Arm rime factors decomosition which is the equivalent of the Taylor formula for decomosition of integers
More informationMATH 361: NUMBER THEORY THIRD LECTURE
MATH 36: NUMBER THEORY THIRD LECTURE. Introduction The toic of this lecture is arithmetic functions and Dirichlet series. By way of introduction, consider Euclid s roof that there exist infinitely many
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 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 informationMaths 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 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 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 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 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 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 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 informationThe 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 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 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 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 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 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 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 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 informationPythagorean 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 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 informationIntroductory Number Theory
Introductory Number Theory Lecture Notes Sudita Mallik May, 208 Contents Introduction. Notation and Terminology.............................2 Prime Numbers.................................. 2 2 Divisibility,
More informationA 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 informationPrimes - 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 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 informationMath 5330 Spring Notes Prime Numbers
Math 5330 Sring 208 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating
More informationPRIME 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 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 informationMATH 250: THE DISTRIBUTION OF PRIMES. ζ(s) = n s,
MATH 50: THE DISTRIBUTION OF PRIMES ROBERT J. LEMKE OLIVER For s R, define the function ζs) by. Euler s work on rimes ζs) = which converges if s > and diverges if s. In fact, though we will not exloit
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 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 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 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 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 informationNumber 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 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 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 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 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
. 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 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 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 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 informationMAS 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 informationMATH 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 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 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 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 informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two oerations defined on them, addition and multilication,
More informationExplicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers
2 3 47 6 23 Journal of Integer Sequences, Vol. 20 207), Article 7.6.4 Exlicit Bounds for the Sum of Recirocals of Pseudorimes and Carmichael Numbers Jonathan Bayless and Paul Kinlaw Husson University College
More information01. Simplest example phenomena
(March, 20) 0. Simlest examle henomena Paul Garrett garrett@math.umn.edu htt://www.math.umn.edu/ garrett/ There are three tyes of objects in lay: rimitive/rimordial (integers, rimes, lattice oints,...)
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 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 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 information6 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 informationPublic Key Cryptosystems RSA
Public Key Crytosystems RSA 57 17 Receiver Sender 41 19 and rime 53 Attacker 47 Public Key Crytosystems RSA Comute numbers n = * 2337 323 57 17 Receiver Sender 41 19 and rime 53 Attacker 2491 47 Public
More informationIdempotent Elements in Quaternion Rings over Z p
International Journal of Algebra, Vol. 6, 01, no. 5, 9-5 Idemotent Elements in Quaternion Rings over Z Michael Aristidou American University of Kuwait Deartment of Science and Engineering P.O. Box 333,
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 informationA LLT-like test for proving the primality of Fermat numbers.
A LLT-like test for roving the rimality of Fermat numbers. Tony Reix (Tony.Reix@laoste.net) First version: 004, 4th of Setember Udated: 005, 9th of October Revised (Inkeri): 009, 8th of December In 876,
More information192 VOLUME 55, NUMBER 5
ON THE -CLASS GROUP OF Q F WHERE F IS A PRIME FIBONACCI NUMBER MOHAMMED TAOUS Abstract Let F be a rime Fibonacci number where > Put k Q F and let k 1 be its Hilbert -class field Denote by k the Hilbert
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 informationOn the Diophantine Equation x 2 = 4q n 4q m + 9
JKAU: Sci., Vol. 1 No. 1, : 135-141 (009 A.D. / 1430 A.H.) On the Diohantine Equation x = 4q n 4q m + 9 Riyadh University for Girls, Riyadh, Saudi Arabia abumuriefah@yahoo.com Abstract. In this aer, we
More informationUniversity of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): 10.
Booker, A. R., & Pomerance, C. (07). Squarefree smooth numbers and Euclidean rime generators. Proceedings of the American Mathematical Society, 45(), 5035-504. htts://doi.org/0.090/roc/3576 Peer reviewed
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 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 information#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 informationLegendre polynomials and Jacobsthal sums
Legendre olynomials and Jacobsthal sums Zhi-Hong Sun( Huaiyin Normal University( htt://www.hytc.edu.cn/xsjl/szh Notation: Z the set of integers, N the set of ositive integers, [x] the greatest integer
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 informationTHE 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 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 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 information#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 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 informationPrime Reciprocal Digit Frequencies and the Euler Zeta Function
Prime Recirocal Digit Frequencies and the Euler Zeta Function Subhash Kak. The digit frequencies for rimes are not all equal. The least significant digit for rimes greater than 5 can only be, 3, 7, or
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 informationAN INTRODUCTION TO GAUSS S NUMBER THEORY. Andrew Granville
AN INTRODUCTION TO GAUSS S NUMBER THEORY Andrew Granville We resent a modern introduction to number theory, aimed both at students who have little exerience of university level mathematics, as well as
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 informationCryptography Assignment 3
Crytograhy Assignment Michael Orlov orlovm@cs.bgu.ac.il) Yanik Gleyzer yanik@cs.bgu.ac.il) Aril 9, 00 Abstract Solution for Assignment. The terms in this assignment are used as defined in [1]. In some
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 informationDIVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
IVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUARATIC FIELS PAUL JENKINS AN KEN ONO Abstract. In a recent aer, Guerzhoy obtained formulas for certain class numbers as -adic limits of traces of singular
More informationAliquot sums of Fibonacci numbers
Aliquot sums of Fibonacci numbers Florian Luca Instituto de Matemáticas Universidad Nacional Autónoma de Méico C.P. 58089, Morelia, Michoacán, Méico fluca@matmor.unam.m Pantelimon Stănică Naval Postgraduate
More informationPrimes 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 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 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 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 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 information