# RANDOM INVOLUTIONS AND THE NUMBER OF PRIME FACTORS OF AN INTEGER 1. INTRODUCTION

Save this PDF as:

Size: px
Start display at page:

Download "RANDOM INVOLUTIONS AND THE NUMBER OF PRIME FACTORS OF AN INTEGER 1. INTRODUCTION"

## Transcription

1 RANDOM INVOLUTIONS AND THE NUMBER OF PRIME FACTORS OF AN INTEGER KIRSTEN WICKELGREN. INTRODUCTION Any positive integer n factors uniquely as n = p e pe pe 3 3 pe d d where p, p,..., p d are distinct prime numbers. We can try to understand the behavior of d as n varies. Mathematicians have thought about this for hundreds of years! An involution is a map f such that composing f with itself gives the identity map ff = id. Let F denote the field with elements, so F = Z/. For each n, there is a beautiful surface called X 0 (n) (which is important for many reasons!) and which looks like the surface of a donut with many holes. The number of these holes is called the genus, and from the surface X 0 (n), one can obtain an involution τ(n) on F g(n), where g(n) denotes the genus of X 0 (n). (Really, one only obtains this involution up to a certain equivalence called isomorphism, but we ll take care of this distinction later.) It is a lovely fact about X 0 (n) that for n odd, there are exactly () g(n)+d elements of F g(n) which are fixed by τ(n). In particular, g(n) and the involution τ(n) determine d. The reason for this is discussed in Section 6, but for now let s think about some consequences. X 0 (n) is called a modular curve and its properties turn out to be extremely important to modern mathematics. The project discussed here is to compute the fixed points of a random involution and compare the number of these to (). In other words:.. Question. Can we model the number of prime factors of an integer by a random involution? A positive answer to this question would be great, but even a negative one might tell us something interesting about X 0 (n). More specifically, τ(n) is the involution induced by complex conjugation on a vector space called H (X 0 (n), F ). It is a fact that τ(n) is determined up to isomorphism by its fixed points (see Theorem 3.6 below), so a negative Date: Summer 0.

2 answer could tell us that τ(n) is not random in a certain sense. Conversely, a positive answer could tell us that τ(n) is random. We need to be more precise about Question. and what the word random means. For instance, for each n, the involution τ(n) is a particular map, and so of course it is not random at all. Let s say what we really mean, and collect some tools to understand involutions and prime factorizations.. HOW BIG IS A RANDOM FINITE SET? Suppose you want to describe all the ways to draw an equilateral triangle of side length in the plane. This can be done by first choosing where to draw the center and then choosing a direction for the line from a vertex to the center. This direction can be any angle from 0 to π, but choosing the angle 0 is no different from choosing the angle π/3, because rotating the triangle π/3 about its center determines an automorphism of the triangle, i.e. a map f from the triangle to itself such that there exists another map g from the triangle to itself with fg and gf equal to the identity map. We can thus describe the ways to draw the triangle as R R/( 3 π). If instead of an equilateral triangle, you draw a triangle whose sides are all of different lengths, the vertices can be distinguished one from another, giving that the ways to draw the triangle forms the space R R/(π). The lack of automorphisms increased the size of the space of possibilities. It is a general principle that one should often count objects weighted by / Aut, where Aut denotes the number of automorphisms. With this principle, we can make sense of the question: How many finite sets are there? Here s how: two objects A and B are said to be isomorphic if there are maps f : A B and g : B A such that the compositions fg and gf are the identity on B and A respectively. Note that any two sets with the same number of elements are isomorphic. So, up to isomorphism, the finite sets are (), {}, {, }, {,, 3}... {,, 3,..., k}.... Note that the automorphisms of the set {,, 3,..., k} are the permutations on k elements, and that there are k! of these. In total, we therefore have k! = e finite sets. k=0.. Example. The probability that a finite set has size k is /(k!e)... Question. How big is a random finite set?

3 .3. Question. Given two finite sets, what is the probability that the larger has at least two more elements that the smaller?.4. Question. Given two finite sets, how much larger do you expect the larger one to be? 3. RANDOM INVOLUTIONS How many F -vector spaces are there? For any positive integer m, F m has the structure of a F -vector space, i.e. you can add two elements of F m together, and you can multiply any element of F m by any element of F. Conversely, any F -vector space can be written as F m for some m. This m is unique, and is called the dimension of the vector space and the function taking an F -vector space to its dimension will be denoted dim. Elements of vector spaces are called vectors, and given vectors v,..., v n, the span of v,..., v n is the set {a v + a v +... a n v n : a i F }. A nice general reference on vector spaces and their properties is [Art9, Ch 3,4]. The automorphisms of F m are the m m invertible matrices, which form GL m (F ). (Multiplication of matrices makes GL m into an object called a group.) The number of elements of GL m (F ) can be computed as follows. The first column can be any non-zero element of F m. There are m of these. After choosing the first n-columns, the n + st column can be any element not in the span of the first n-columns. So there are m choices for the second column, and more generally, m n choices for the n+st column. Thus m GL m F = ( m n ) [Lan0, XIII Ex 5 p 546] and there are F -vector spaces. m m= n= n= m n 3.. Question. What s the probability that an F -vector space has dimension 3? What s the probability that an F -vector space has dimension 5? Compare the sizes of these probabilities e.g. how many times more likely is it that an F -vector space has dimension 3? Is it more likely that an F -vector space has dimension 00 or 0? How many F -vector spaces with involutions are there? To compute the number of F -vector spaces with involution, we need two things: () A list of the isomorphism classes of F -vector spaces with involution. 3

5 that m+0 = m for all m M, an element m such that m+( m) = 0, and multiplication by elements of R distributes over addition, i.e. r(m +m ) = rm +rm for all m, m M and r R. A reference on modules is [Art9, Ch ] The relationship between involutions and F [Z/]-modules. An involution f on an F - vector space F n gives Fn the structure of an F [Z/]-module by defining for all (a + a τ) F [Z/] and all v in F n, (a + a τ)v = a v + a fv F n. Conversely an F [Z/]-module is an F -vector space with an involution, by letting the vectors of the vector space be the elements of the module and setting the involution equal to τ. Two important examples of F [Z/]-modules are F [Z/] itself, and F. In the former case, the underlying vector space is F and the involution f is ( ) 0 f =. 0 In the latter case, the underlying vector space is F and the involution is the identity Exercise. Write down a matrix corresponding to the involution on F a+b given by the F [Z/]-module F [Z/] a F b. For starters, take a = and b =. for a unique pair of non Theorem. Any F [Z/]-module is isomorphic to F [Z/] a F b negative integers (a, b). Since the pair (a, b) is unique, we can consider a and b as functions which take an F [Z/] module and return a non-negative integer. We ll adopt this convention, although it s not standard. Note that the dimension of an F [Z/]-module is dim = a + b. Theorem 3.6 is true because the ring F [Z/] is an example of a principal ideal domain. See [Lan0, III 7 p. 46] if you are interested. Theorem 3.6 says that the isomorphism classes of involutions on F n are in bijection with pairs of non-negative integers (a, b) such that a + b = n. In particular, there are n/ of these given by F n, F [Z/] F n, F [Z/] F n Exercise. Let n be an odd integer with d distinct prime factors. Recall that τ(n) is an involution on F g(n) with exactly g(n)+d fixed points. Show that τ(n) is the involution associated to the F [Z/]-module F [Z/] a F b for a = g(n) d + and b = ( d ). We now wish to understand the automorphisms of F [Z/] a F b. An automorphism T : F [Z/] a F b F [Z/] a F b 5

6 is given by an (a + b) (a + b) matrix, separated into blocks of size a a, a b, b a and b b T = with e ij, f ij in F [Z/], and c ij, d ij in F. e e... e a f... f b e e... e a f... f b.... e a e a... e aa f a... f ab c c... c a d... d b.... c b c b... c ba d b... d bb By Example 3.3, we can replace F [Z/] with F [X]/ X. Since X acts by 0 on F, it follows that Xf ij = 0, which implies that f ij = Xf ij for some f ij F. T is an automorphism exactly when the determinant of T in F [X]/ X has its constant coefficient equal to, i.e. Det T = +a X for a F. Viewing Det T as a polynomial in the variable X, we can write this as follows: T is an automorphism if and only if (Det T)(0) =. Since the determinant is a polynomial in the matrix entries, we are free to consider the matrix entries as polynomials in X and evaluate at 0 before taking the determinant. Thus, f ij can be any element of F for all i, j and the coefficient of X in all the e ij can also vary freely. Since the determinant of [ E 0 C D is Det E Det D, we have that the automorphisms of F [X]/ X a F b are in bijection with the matrices [ ] E + E X F X C D with E GL a F, D GL b F, C Mat b a F, E Mat a F, and F Mat a b F. Thus: ] 3.8. Proposition. Let (a, b) be a pair of non-negative integers. The F [Z/]-module F [Z/] a F b has exactly GL a F GL b F Mat b a F Mat a a automorphisms Exercise. Express the number of automorphisms of F [Z/] a F b a and b. as a polynomial in 3.0. Exercise. Let (a, b) be a pair of non-negative integers such that a + b = n. What is the probability that an involution on F n is isomorphic to the involution corresponding to F [Z/] a F b? 6

7 3.. Remark. There are finitely many involutions on F n. Call the number of these D. Some finite subset of these will correspond to an F n -module isomorphic to F [Z/] a F b. Call the number of these N. The quotient N/D is equal to / Aut(a, b) a,b / Aut(a, b ) where Aut(a, b ) denotes the number of automorphisms of the F [Z/]-module F [Z/] a F b and the sum is taken over pairs of non-negative integers (a, b ) such that a +b = n. 3.. Exercise. What is the expected number of fixed points of an involution on F n? 3.3. Exercise. How many F -vector spaces with involution are there? 3.4. Exercise. Fix a positive integer n. What is the probability that a randomly chosen F -vector space with involution will have dimension n? In other words, what is the probability that a random F -vector space with involution is isomorphic to some F [Z/] a F b with a + b = n? 3.5. Exercise. Fix a positive integer N. What is the expected value of a dim for a random involution on a vector space of dimension > N? Note that taking N = gives the expected value of a dim for a random involution. The expected value of a function is the sum over the possibilities of the probability of the possibility times the value of the function i.e. Expectedvalue(f) = Probability(a)f(a). possibilities a 4. SOME INFORMATION ON THE DISTRIBUTION OF PRIMES Mertens nd theorem states that (3) p n p log log n M 4 log(n + ) + n log n where log denotes the logarithm base e and M denote s the Meissel-Mertens constant which is about In particular, lim ( log log n) = M. n p p n Let d(n) denote the number of distinct prime factors of n. Let π(n) denote the number of primes n. We approximate M n=n+ d(n). For a prime p M, the number of n in 7

8 [N +, M] which are divisible by p is (M N)/p + r p, where r [0, ). Thus (4) (M N) p M M p n=n d(n) (M N) p M p + π(m) 4.. Exercise. Combine (4) and (3) to obtain bounds for M n=n d(n) that do not contain sums over all primes M. 4.. Exercise. Obtain bounds for M n>n d(n) where n only runs over the odd integers contained in (N, M]. π(n) n/ log n n The Prime Number Theorem says that lim n =. (The difference π(n) is log n related to the Riemann hypothesis. The Riemann hypothesis is perhaps the most famous unsolved problem in math.) More generally, let π(n, k) denote the number of positive integers n which such that d = k. Let Then G k (n) = n log n (log log n) k. (k )! π(n, k) lim n G k (n) =. The Hardy-Ramanujan inequality gives that there exist constants c and c such that for all n 3 and all k. n (log π(n, k) c n + c ) k log n (k )! See [HT88] for more information on π(n, k). 5. PROJECT DESCRIPTION The goal is to compare random involutions with τ(n) for a lot of odd values of n. The genus g(n) of X 0 (n) has a rather involved formula, which is given below as (5). However, the dominant term should be n. Consider the following model or approximation of τ(n). 5.. Model. τ(n) is similar to a random involution on an F -vector space of dimension n. 5.. Remark. A random involution is much more likely to have small dimension than large dimension (see Exercise 3.4.) However g(n) gets arbitrarily large as n gets large. 8

9 Model 5. is forcing the random involution to have its underlying vector space be reasonably large Project. Test 5. as follows. Recall that the value of a dim for τ(n) is d + for n odd by Exercise 3.7. Recall as well that we have estimates for M n>n d(n) where n runs over the odd integers from N + to M by Exercise 4.. For n odd N < n M, compute the expected value of a dim of a random involution of dimension n, and call the result E(n). This was Exercise 3.5. Compute S(N, M) = M N<n log ( E(n)), where this sum runs over odd n. Compare S(N, M) to the estimates obtained for M n>n d(n) in Exercise 4. by giving bounds for S(N, M) M n>n d(n). Can you compute the limit lim lim N M S(N, M) M n>n d(n)? Or lim N S(N, N) N n>n d(n)? We can also try to be more precise about g(n). The genus g(n) of X 0 (n) is given by the formula: (5) g(n) = + µ ν 4 ν 3 3 ν where for n odd µ = n p n ν = p n { ν 3 = ν = d n ( + p ) ( + ( p )) 0 if 9 n, p n ( + ( 3 p ϕ((d, n/d)) )) otherwise. Here ϕ is Euler s function, ϕ() =, and ( p ) is the quadratic residue symbol, so 9

10 { ( p ) = if p mod (4), if p 3 mod (4), ( 3 p ) = 0 if p = 3, if p mod (3), if p mod (3), See [Shi7, Prop.40, Prop.43] Refined project. Find lower bounds for g(n) and replace n How does S(N, M) change as you adjust these bounds? by these bounds in 5.3. It would be nice to know if a randomly chosen τ(n) is similar to a randomly chosen involution. This seems hard to me, but here are possible projects along these lines: 5.5. Project. Choose a random F [Z/] module. Let E be the expected value of a dim. Choose a random non-negative integer, where random means that the probability of choosing k is /(k!e). Try to compute the expected number E of prime factors d of k. Compare E and E +. Interpret the comparison using Exercise Project. Choose a random non-negative integer, where random means that the probability of choosing k is /(k!e). Try to compute the expected number of prime factors d of k. Try to compute the expected value of g(k)+d(k). Compare this to the expected number of fixed points of a random F -vector space with involution. There is an operation called direct sum on modules and denoted by. In fact, taking the direct sum of finitely many modules is the same as taking their product. It would be nice if the involution τ(n, M) = M n=n+ τ(n) on M n=n+ Fg(n) could be modeled by random involutions. This again seems hard to me, but hopefully I m wrong, so here is a project about this. (Note that since these direct sums are finite, you are welcome to replace with the product. The reason for this notation is that if we let M, the product and direct sum will be different, and I think is more appropriate, because it would correspond to H of X 0 (n).) 5.7. Project. If possible, compare τ(n, M) with the direct sum of randomly chosen involutions on F g(n) as n ranges over odd integers between N + and M. 6. NOTES ON THE PROJECT It is a result of Ogg, Akbas, and Singerman that for n odd, the real points of X 0 (n) form exactly d circles, where d is the number of distinct prime factors of n. See [AS9, p. 6] [Sno, Prop..]. (The reason for this result is that for each primary part of the marked 0

11 cyclic subgroup of an elliptic curve with level structure, complex conjugation can either act trivially or not.) It follows that H (X 0 (n), Z) = Z[Z/] a Z() b Z b, with b = d, for instance by [Wic, Prop 3.]. Thus a = g(n) d +. As pointed out to me by Andrew Snowden, if the real structure on X 0 (n) is twisted in a certain way, the number of connected components of real points is related to the size of the class groups of certain number fields. So randomness in those class groups is related to randomness in the corresponding involution. It s fun to recall the Cohen- Lenstra heuristics in this context. This project is partially based on joint work with Andrew Snowden. REFERENCES [Art9] Michael Artin, Algebra, Prentice Hall Inc., Englewood Cliffs, NJ, 99. MR 9886 (9g:0000) [AS9] M. Akbas and D. Singerman, Symmetries of modular surfaces, Discrete groups and geometry (Birmingham, 99), London Math. Soc. Lecture Note Ser., vol. 73, Cambridge Univ. Press, Cambridge, 99, pp. 9. MR 9690 (93j:0099) [HT88] Adolf Hildebrand and Gérald Tenenbaum, On the number of prime factors of an integer, Duke Math. J. 56 (988), no. 3, MR (89k:084) [Lan0] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol., Springer-Verlag, New York, [Shi7] 00. MR (003e:00003) Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No.. Iwanami Shoten, Publishers, Tokyo, 97, Kanô Memorial Lectures, No.. MR (47 #338) [Sno] Andrew Snowden, Real components of modular curves, arxiv:08.33v, 0. [Wic] Kirsten Wickelgren, -nilpotent real section conjecture, arxiv: v, 0. address:

### 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

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Group Theory (MA343): Lecture Notes Semester I Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway

Group Theory (MA343): Lecture Notes Semester I 2013-2014 Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics, NUI Galway November 21, 2013 Contents 1 What is a group? 2 1.1 Examples...........................................

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 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

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### 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

### 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(

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### = 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

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

### 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

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### 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

### MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

### EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

### 1 Symmetries of regular polyhedra

1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### MATH 361: NUMBER THEORY FIRST LECTURE

MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,

### 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

### Computing the Symmetry Groups of the Platonic Solids With the Help of Maple

Computing the Symmetry Groups of the Platonic Solids With the Help of Maple Patrick J. Morandi Department of Mathematical Sciences New Mexico State University Las Cruces NM 88003 USA pmorandi@nmsu.edu

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Short Programs for functions on Curves

Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.

### G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE

Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 0019-2082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Integer Sequences and Matrices Over Finite Fields

Integer Sequences and Matrices Over Finite Fields arxiv:math/0606056v [mathco] 2 Jun 2006 Kent E Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA 93407 kmorriso@calpolyedu

### 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

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 4 1. SCHOOF S ALGORITHM Let K be a finite field with q elements. Let p be its characteristic. Let X be an elliptic curve over K. To simplify we assume

### ABSTRACT ALGEBRA. Romyar Sharifi

ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory

### 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

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### 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

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 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

### Pre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems

Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small

### Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Examples of Functions

Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### 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

### CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

### GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Algebraic and Transcendental Numbers

Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### 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

### SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

### Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005

Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 A.J. Hildebrand Department of Mathematics University of Illinois http://www.math.uiuc.edu/~hildebr/ant Version 2013.01.07 18 Chapter

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Galois representations with open image

Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### Algebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Factoring of Prime Ideals in Extensions

Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### 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

### Prime numbers and prime polynomials. Paul Pollack Dartmouth College

Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)

### ECE 842 Report Implementation of Elliptic Curve Cryptography

ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### Cyclotomic Extensions

Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication