# Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Save this PDF as:

Size: px
Start display at page:

Download "Every Positive Integer is the Sum of Four Squares! (and other exciting problems)"

## Transcription

1 Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer is the sum of four squares. For instance, 5 = , 1 = , 17 = Proof. We will use the following Euler s identity: (x 1 + x + x 3 + x 4)(y 1 + y + y 3 + y 4) = (x 1 y 1 + x y + x 3 y 3 + x 4 y 4 ) +(x 1 y x y 1 +x 3 y 4 x 4 y 3 ) +(x 1 y 3 x 3 y 1 +x 4 y x y 4 ) +(x 1 y 4 x 4 y 1 +x y 3 x 3 y ) which is very easy to verify. The conclusion is that the product of two numbers that are sum of four squares is also sum of four squares. Hence, since the case of 1 is trivial, and since every natural number > 1 can be decomposed as a product of prime numbers, it is enough to prove the result for prime numbers. The first case is p =, but that follows from = For the case of p odd, we are going to need the following: Lemma If p is an odd prime, then there are numbers x, y, and m such that 1 + x + y = mp 0 < m < p So, for instance, for p = 3 we have = 3, for p = 7 we have = 3 7. Proof. For x = 0, 1,..., p 1, the numbers x have all of them different congruences modulo p. This is because if x 1 x (modp), then p (x 1 x )(x 1 + x ) x 1 ±x (modp) which is a contradiction. So we have p+1 numbers which are incongruent modulo p. For y = 0, 1,..., p 1, the numbers 1 y are all incongruent modulo p, using the same idea as before. So we have another set of p+1 numbers incongruent modulo p. But there are p + 1 numbers altogether in these two sets, and only p possible residues modulo p. Then at least one number x in the first set must be congruent to a number 1 y in the second set. Hence, x 1 y (modp) x y = mp Now x < ( p) and y < ( p ) so, ( p ) mp = 1 + x + y < 1 + < p 1

2 then m < p. It follows from the Lemma that, for p odd prime, there is an 0 < m < p such that mp = x 1 + x + x 3 + x 4 We are going to see that the least m with that property is m = 1. Let m 0 be the least m with the property. If m 0 = 1 there is nothing to prove. Then we can suppose 1 < m 0 < p. If m 0 is even, then either the x i are all even, or the x i are all odd, or two of them are even and the other two are odd. In this case, say x 1, x are even. Then in all the three cases, x 1 ± x and x 3 ± x 4 are all even and we can write m 0 p = ( ) x1 + x ( ) x1 x ( ) x3 + x ( 4 x3 x and this contradicts the minimality of m 0. Now we choose y i such that y i x i (modm 0 ), y i < m 0 This can be done, since m 0 1 y m 0 1 is a complete set of residues. Now observe that the x i are not all divisible by m 0, since this would imply m 0 m 0 p and m 0 p, a contradiction. As a consequence, we get Then Therefore, and so, y 1 + y + y 3 + y 4 > 0 y 1 + y + y 3 + y 4 < m 0 and y 1 + y + y 3 + y 4 0 (mod m 0 ) x 1 + x + x 3 + x 4 = m 0 p (m 0 < p) y 1 + y + y 3 + y 4 = m 0 m 1 (0 < m 1 < m 0 ) m 1 m 0 p = z 1 + z + z 3 + z 4 where the z i come from Euler s identity. But z 1 = x 1 y 1 + x y + x 3 y 3 + x 4 y 4 x 1 + x + x 3 + x 4 0 (mod m 0) and similarly the other z i are also divisible by m 0. We can write then z i = m 0 w i. Dividing by m 0, we get m 1 p = w 1 + w + w 3 + w 4 and this contradicts again the minimality of m 0. It follows that m 0 = 1. We should also mention that it is possible to compute the number of such representations. We will state without prove the following Theorem 3 Let Q(n) the number of solutions of ) n = x 1 + x + x 3 + x 4

3 then, writing n = r (t + 1) Q(n) = { 8 S(t + 1) = 8S(n) forr = 0 4 S(t + 1) forr 0 where S(n) = d n d We have seen that four squares are enough to represent any natural number. Three squares are not enough. Indeed, x i 0, 1 or4(mod 8), therefore, x 1 + x + x 3 7 (mod 8) hence no number of the form 8t + 1 can be represented by three squares. We have the following Theorem (which we won t prove here). Theorem 4 Let n be a positive integer. Then n can be expressed as the sum of three squares if and only if n is not of the form 4 r (8t + 7). For the case of two squares, Theorem 5 Let n be a positive integer. Then n can be expressed as the sum of two squares if and only if all prime factors of n of the form 4t+3 have even exponents in the factorization of n. This Theorem can be proved in a similar way as we did for four squares, using the identity (x 1 + x )(y 1 + y ) = (x 1 y 1 + x y ) + (x 1 y x y 1 ). Waring s problem Waring s problem has to do with representing natural numbers as the sum of a fixed number s of k-th powers, namely, n = x k x k s If we fix k > 1 and s is too small, the problem cannot be solved for every n. Say that s = 1, then we won t get solutions unless n is a k-th power. The first arising question is whether, for a given k, there is any s = s(k) such that n = x k x k s is soluble for every n. In 1770, Waring stated that every number is expressible as a sum of 4 squares, 9 cubes, 19 biquadrates, and so on, implying that s does exist. Hilbert was the first to prove this assertion for every k, in Clearly, given an s that works, any s > s will work the same. Hence there must be a minimal s that works. It is ussually denoted by g(k). We have just proved that g() = 4. It is not hard to prove 3

4 Theorem 6 g(4) does exist and it is 53. Proof. Let us denote by B s a number which is the sum of s biquadrates. The identity 6(a + b + c + d ) = (a + b) 4 + (a b) 4 + (c + d) 4 + (c d) 4 + (a + c) 4 + (a c) 4 + (b + d) 4 + (b d) 4 + (a + d) 4 + (a d) 4 + (b + c) 4 + (b c) 4 shows us that 6(a + b + c + d ) = B 1 Because of Lagrange s Theorem, we get 6x = B 1 for every x. Now every positive number is of the form n = 6t + r with 0 r 5, then using Lagrange s Theorem once again, implying Hence g(4) exists and it is at most 53. n = 6(x 1 + x + x 3 + x 4) + r n = B 1 + B 1 + B 1 + B 1 + r = B 48 + r = B 53 Theorem 7 [ (3 ) ] k g(k) k + Proof. Write q = [ (3 ) k ]. Consider the number n = k q 1 < 3 k which can be only represented by terms of the form 1 k and k. Since, n = (q 1) k + ( k 1)1 k n requires k + q powers. It has been proved that g(k) = k +q for every but finitely many k. It is conjectured that this equality is always true. There is another number that in a sense is more interesting that g(k). We define G(k) as the least value of s for which it is true that every positive integer which is large enough can be expressed as a sum of s k-th powers. Thus G(k) g(k) For k =, we get G() = 4 since we have proved that four squares are enough and we have exhibited infinitely many numbers, the ones of the form 8t+7, that cannot we writen as sum of only three squares. 4

5 In general G(k) is much smaller than g(k). Take k = 3. It is known that g(3) = 9. Every number can be represented by the sum of nine cubes. Indeed every number but 3 = and 39 = can be expressed by the sum of at most eight cubes. Moreover, there are only 15 integers that require 8 cubes. This implies that G(3) 7. G(k) is only known for the cases k =, 4. The best currently known bound for G(k) is for some constant c. On the other hand, Theorem 8 for k G(k) < c k log k G(k) k + 1 Proof. Let A(N) be the number of n N which are representable as with x i 0. We may suppose n = x k x k k 0 x 1 x... x k N 1 k If B(N) is the number of solutions of the inequality, then A(N) B(N). Clearly, ( 1 ) ( ) k [N k ] k 1 i=1 [N 1 k ] + i B(N) = = N k k! k! If G(k) k, all but finitely many numbers are representable as the sum of k kth powers and A(N) N c where c is just a constant (independent of N). Then, N c A(N) B(N) N k! and this is impossible when k > 1. Finally, the Table shows what is known for the first 0 values of k. 5

6 k g(k) G(k) References [1] H. Davenport, Analytic Methods for Diophantine Equations and Diophantine inequalities, The University of Michigan, Fall Semester (196) [] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press, Oxford (1998) [3] E. Landau, Elementary Number Theory, Chelsea Publishing Company, New York (1958) [4] Eric Weisstein s World of Mathematics, 6

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Doug Ravenel. October 15, 2008

Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we

### Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### Primality - Factorization

Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

### TEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011

TEXAS A&M UNIVERSITY Prime Factorization A History and Discussion Jason R. Prince April 4, 2011 Introduction In this paper we will discuss prime factorization, in particular we will look at some of the

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### PROOFS BY DESCENT KEITH CONRAD

PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the

### Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

### An Introductory Course in Elementary Number Theory. Wissam Raji

An Introductory Course in Elementary Number Theory Wissam Raji 2 Preface These notes serve as course notes for an undergraduate course in number theory. Most if not all universities worldwide offer introductory

### NON-CANONICAL EXTENSIONS OF ERDŐS-GINZBURG-ZIV THEOREM 1

NON-CANONICAL EXTENSIONS OF ERDŐS-GINZBURG-ZIV THEOREM 1 R. Thangadurai Stat-Math Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700035, INDIA thanga v@isical.ac.in Received: 11/28/01,

### Pythagorean vectors and their companions. Lattice Cubes

Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived

### Die ganzen zahlen hat Gott gemacht

Die ganzen zahlen hat Gott gemacht Polynomials with integer values B.Sury A quote attributed to the famous mathematician L.Kronecker is Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.

### On the number-theoretic functions ν(n) and Ω(n)

ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### Factorization Algorithms for Polynomials over Finite Fields

Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 2011-05-03 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is

### The cyclotomic polynomials

The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =

### QUADRATIC RECIPROCITY IN CHARACTERISTIC 2

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### Factoring & Primality

Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### CIS 5371 Cryptography. 8. Encryption --

CIS 5371 Cryptography p y 8. Encryption -- Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: All-or-nothing secrecy.

### Monogenic Fields and Power Bases Michael Decker 12/07/07

Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Z-module, the nicest possible case is

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Cryptography and Network Security Chapter 8

Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:

### On prime-order elliptic curves with embedding degrees k = 3, 4 and 6

On prime-order elliptic curves with embedding degrees k = 3, 4 and 6 Koray Karabina and Edlyn Teske University of Waterloo ANTS VIII, Banff, May 20, 2008 K. Karabina and E. Teske (UW) Prime-order elliptic

### Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

### Lecture 13: Factoring Integers

CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method

### Jacobi s four squares identity Martin Klazar

Jacobi s four squares identity Martin Klazar (lecture on the 7-th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

### Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

### FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Congruence properties of binary partition functions

Congruence properties of binary partition functions Katherine Anders, Melissa Dennison, Jennifer Weber Lansing and Bruce Reznick Abstract. Let A be a finite subset of N containing 0, and let f(n) denote

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### Prime power degree representations of the symmetric and alternating groups

Prime power degree representations of the symmetric and alternating groups Antal Balog, Christine Bessenrodt, Jørn B. Olsson, Ken Ono Appearing in the Journal of the London Mathematical Society 1 Introduction

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY

FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

### DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

### Algebra & Number Theory. A. Baker

Algebra & Number Theory [0/0/2009] A. Baker Department of Mathematics, University of Glasgow. E-mail address: a.baker@maths.gla.ac.uk URL: http://www.maths.gla.ac.uk/ ajb Contents Chapter. Basic Number

### Basic Algorithms In Computer Algebra

Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### 4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

### On some Formulae for Ramanujan s tau Function

Revista Colombiana de Matemáticas Volumen 44(2010)2, páginas 103-112 On some Formulae for Ramanujan s tau Function Sobre algunas fórmulas para la función tau de Ramanujan Luis H. Gallardo University of

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Powers of Two in Generalized Fibonacci Sequences

Revista Colombiana de Matemáticas Volumen 462012)1, páginas 67-79 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian

### THE CONGRUENT NUMBER PROBLEM

THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean

### FACTORING SPARSE POLYNOMIALS

FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S

### LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO

LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO PETER MÜLLER AND MICHAEL E. ZIEVE Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects.

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### FACTORING. n = 2 25 + 1. fall in the arithmetic sequence

FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard,

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Some applications of LLL

Some applications of LLL a. Factorization of polynomials As the title Factoring polynomials with rational coefficients of the original paper in which the LLL algorithm was first published (Mathematische

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### Bounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances

Bounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances Jialin Zhang Tsinghua University zhanggl02@mails.tsinghua.edu.cn Wei Chen Microsoft Research Asia weic@microsoft.com Abstract

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### A Hajós type result on factoring finite abelian groups by subsets II

Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### Primality Testing and Factorization Methods

Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,

### THE SCHEDULING OF MAINTENANCE SERVICE

THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GUSTAVO A. FERNÁNDEZ-ALCOBER AND ALEXANDER MORETÓ Abstract. We study the finite groups G for which the set cd(g) of irreducible complex

### An Overview of Integer Factoring Algorithms. The Problem

An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm

### Abstract Algebra Theory and Applications. Thomas W. Judson Stephen F. Austin State University

Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University August 16, 2013 ii Copyright 1997-2013 by Thomas W. Judson. Permission is granted to copy, distribute and/or

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### r + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn.

Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in