# MATH 361: NUMBER THEORY FIRST LECTURE

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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, consider the prime numbers, the noninvertible positive integers divisible only by 1 and by themselves, P = {2, 3, 5, 7, 11, }. Euclid (c.300 B.C.) and Diophantus (c.250 A.D.) posed and solved number theory problems, as did Archimedes. In the middle ages the Indians and perhaps the Chinese knew further results. Fermat (early 1600 s) got a copy of Diophantus and revived the subject. Euler (mid-1700 s), Lagrange, Legendre, and others took it seriously. Gauss wrote Disquisitiones Arithmeticae in Example questions: (The perfect number problem.) The numbers 6 = and 28 = are perfect numbers, the sum of their proper divisors. Are there others? Which numbers are perfect? This problem is not completely solved. (The congruent number problem.) Consider a positive integer n. Is there a rational right triangle of area n? That is, is there a right triangle all three of whose sides are rational, that has area n? This problem is not completely solved, but very technical 20th century mathematics has reduced it to a problem called the Birch and Swinnerton-Dyer Conjecture, one of the socalled Clay Institute Millennium Problems, each of which carries a million dollar prize. The book Introduction to Elliptic Curves and Modular forms by Neal Koblitz uses the congruent number problem to introduce the relevant 20th century mathematics. (A representation problem.) Let n be a positive integer. Which primes p take the form p = x 2 + ny 2 for some x, y Z? This problem is solved by modern mathematical ideas, specifically complex multiplication and class field theory. The book Primes of the Form x 2 + ny 2 by David Cox uses the representation problem in its title to introduce these subjects. (Questions about the distribution of primes.) Are there infinitely many primes? Infinitely many 4k + 1 primes? Infinitely many 28k + 9 primes? If so, can we say more than infinitely many? If π(x) denotes the number of primes p x then how does π(x) grow as x grows? Questions like this lead quickly into analytic number theory, where, for example, ideas from 1

2 2 MATH 361: NUMBER THEORY FIRST LECTURE complex analysis or Fourier analysis are brought to bear on number theory. It is known that asymptotically π(x) x/ ln(x) but the rate of asymptotic convergence depends on the famous Riemann hypothesis, another unsolved problem. (The Goldbach Conjecture.) Is every even integer n 4 the sum of two primes? This problem is unsolved. For all of these problems, the answer is either unknown or requires mathematical structures larger than Z + and its arithmetic. On the other hand, plenty can be done working entirely inside Z + as well. So, loosely speaking, there are two options for a first course in number theory, elementary or nonelementary. (Here elementary doesn t mean easy, but rather refers to a course set entirely in Z.) And again speaking loosely, a nonelementary course can be algebraic or analytic. This course will be nonelementary, with more emphasis on algebra than on analysis, although I hope to introduce some analytical ideas near the end of the semester to demonstrate their interaction with the algebra. Despite the emphasis on algebra in this class, the abstract algebra course is not prerequisite. This course is equally a good venue for practicing with algebra or for beginning to learn it. Our text, by Ireland and Rosen, is well-suited to the emphasis of the course. We will cover its first nine chapters and a selection of its later material. Many books are on reserve for this course as well, e.g., Cox, Hardy and Wright, Koblitz, Marcus, Silverman and Tate, Niven and Zuckerman and Montgomery, and so on. Many of these books are mostly about subjects that we will only touch on. Feel welcome to come see me for guidance about reading beyond the course. 2. The Gaussian Integers As an example of an algebraic structure larger than the integers, the ring of Gaussian integers is Z[i] = {a + ib : a, b Z}, with its rules of addition and multiplication inherited from the field of complex numbers. The Gaussian integers form a ring rather than a field, meaning that addition, subtraction, and multiplication are well-behaved, but inversion is not: the reciprocal of a Gaussian integer in general need not exist within the Gaussian integers. As a ring, the Gaussian integers behave similarly to the rational integers Z. The units (multiplicatively invertible elements) of the Gaussian integers are Z[i] = {±1, ±i}, a multiplicative group. Every nonzero Gaussian integer factors uniquely (up to units) into prime Gaussian integers. And so on. Prime numbers in Z are called rational primes to distinguish them from prime numbers in the Gaussian integers. The somewhat awkward phrase odd prime means any prime p other than 2.

3 MATH 361: NUMBER THEORY FIRST LECTURE 3 3. Prime Sums of Two Squares via the Gaussian Integers Theorem 3.1 (Prime Sums of Two Squares). An odd rational prime p takes the form p = a 2 + b 2 (where a, b Z) if and only if p 1 mod 4. (Note: The notation p 1 mod 4 means that p = 4k + 1 for some k. In general, the language x is y modulo n means that x and y have the same remainder upon division by n, or equivalently, that n divides y x.) Proof. ( = ) This direction is elementary. An odd rational prime p is 1 or 3 modulo 4. If p = a 2 + b 2 then p 1 mod 4 because each of a 2 and b 2 is 0 or 1 modulo 4. ( = ) This direction uses the Gaussian integers. For now, take for granted a fact about their arithmetic: Granting the fact, we have p 1 mod 4 = p factors in Z[i]. p 1 mod 4 = p factors in Z[i] = p = (a + ib)(c + id), a + ib, c + id / Z[i] = p 2 = pp = (a 2 + b 2 )(c 2 + d 2 ) = p = a 2 + b 2 = c 2 + d 2. So the arithmetic of the Gaussian integers has made the problem easy, but now we need to establish their arithmetic property that p factors in Z[i] if p 1 mod 4. To do so structurally, we quote a fact to be shown later in this course, {1, 2, 3,, p 1} = {1, g, g 2,, g p 2 } for some g, working modulo p. That is, some element g of the multiplicative group {1, 2, 3,, p 1} (again, working modulo p) generates the group. Equivalently, the group is cyclic. The generator g need not be unique, but choose some g that generates the group, and let h = g (p 1)/4. The definition of h is sensible since p 1 mod 4. Then h 2 = 1, still working modulo p. That is, in the mod p world, 1 has a square root. Now work in the ordinary integers again, where we have h 2 1 mod p. That is, letting the symbol mean divides (and then moving back up to the Gaussian integers), p h = (h + i)(h i) But if p h + i then also p = p h i, so p 2i, so p 2 in Z[i], so p 2 in Z, contradicting the fact that p is odd. Similarly, p h i. We conclude that p is not prime in Z[i], because when a prime divides a product, it divides at least one of the factors. Thus p factors in Z[i] as desired. However, this argument leaves us with a loose end. Does prime mean doesn t decompose as a product (i.e., divisible only by itself, up to units), or does prime mean doesn t decompose as a factor (i.e., if it divides a product then it must divide at least one multiplicand)? We will discuss this issue soon.

4 4 MATH 361: NUMBER THEORY FIRST LECTURE 4. Pythagorean Triples via the Gaussian Integers Suppose that we have a primitive Pythagorean triple, x 2 + y 2 = z 2, x, y, z Z +, gcd(x, y, z) = 1. It follows that in fact x, y, and z are pairwise coprime. We normalize the triple by taking x odd, y even, and z odd. (Inspection modulo 4, as in the beginning of the proof just given, shows that the case where x and y are odd and z is even can not arise.) Working in the Gaussian integers, the sum of squares factors, z 2 = (x + iy)(x iy). For now, take for granted the fact that consequently Granting the fact, we have x + iy is a perfect square in Z[i]. x + iy = (r + is) 2 = r 2 s 2 + i2rs, so that the primitive normalized Pythagorean triple takes the form where x = r 2 s 2, y = 2rs, z = r 2 + s 2, 0 < s < r, gcd(r, s) = 1, one of r, s is even. Thus we can systematically write down all primitive normalized Pythagorean triples in a table. The table begins as follows. r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 s = 1 (3, 4, 5) (15, 8, 17) (35, 12, 37) s = 2 (5, 12, 13) (21, 20, 29) (45, 28, 53) s = 3 (7, 24, 25) (27, 36, 45) s = 4 (9, 40, 41) (33, 56, 65) s = 5 (11, 60, 61) s = 6 (13, 84, 85) So the question is why x + iy is a perfect square in Z[i]. For any prime π Z[i] we have the implications π x + iy = π (x + iy)(x iy) = z 2 = π 2 z 2 (by unique factorization) = π 2 x + iy or π x iy (since already π x + iy). If π 2 x + iy then the prime π occurs at least twice in x + iy, so divide x + iy by π 2 and repeat the argument. The only other scenario is π x + iy and π x iy. But { } π x + iy { } π 2x { } ππ 4x 2 π x iy = π z = ππ z 2 The last two paired conditions hold in Z[i] if and only if they hold in Z. They fail in Z because of the conditions on the primitive normalized Pythagorean triple (x, y, z). So the condition π x iy is impossible, and the argument is complete.

5 MATH 361: NUMBER THEORY FIRST LECTURE 5 It is natural to wonder whether the same idea of factoring in a suitable ring might help with Fermat s Last Theorem. Let p be an odd prime, and suppose that we have a Fermat triple x p + y p = z p. Then p 1 z p = x p + y p = (x + ζpy), j ζ p = e 2πi/p. j=0 (The factorization just given is perhaps most easily seen by noting that x p + y p = x p ( y) p since p is odd, and so x p + y p is 0 exactly when x = ζ j p( y) = ζ j py for some j {0, 1,, p 1}.) For the argument to continue along the lines of the argument for Pythagorean triples, we need to know that elements factor uniquely into primes (up to units) in the ring Z[ζ p ]. Unique factorization holds for p = 2, 3, 5, 7, 11, 13, 17, 19. But unique factorization fails for p = 23, and it fails in general. 5. Rational Parametrization of Conic Curves Let k denote any field, and let K be any extension field of k, possibly K = k. A line defined over k is an equation L : Ax + By = C, A, B, C k, where at least one of A, B is nonzero. A K-rational point of L is a solution (x, y) K 2 of L. The set of K-rational points of L is denoted L K. Similarly, a conic curve defined over k is an equation C : ax 2 + bxy + cy 2 + dx + ey + f = 0, a, b, c, d, e, f k, where at least one of a, b, c nonzero. A K-rational point of C is a solution (x, y) K 2 of C, and the set of K-rational points of C is denoted C K. (Note: C K may not contain any points at all. For example, let k = K = R and consider the conic curve C : x 2 + y 2 = 1.) Proposition 5.1. Suppose that C K contains a point P = (x P, y P ) not in L K. Then the points of C K other than P are in bijective correspondence with the points of L K. Proof. First note that after a coordinate translation, we may let P = (0, 0), although now the coefficients of L and C could lie in K rather than k. For any given point Q = (x Q, y Q ) C K such that Q P, let t = y Q /x Q K and then solve the equation L(x, tx) for a unique x R K. Let y R = tx R. The point R = (x R, y R ) L K is collinear with P and Q. Conversely, for any given point R = (x R, y R ) L K such that R P, let t = y R /x R K and then consider the equation C(x, tx). This quadratic equation has x = 0 as a solution, but after dividing the equation through by x there is a unique second solution x Q K. (Possibly x Q = 0 as well.) Let y Q = tx Q. The point Q = (x Q, y Q ) C K is collinear with P and R. The argument here has left out the case where all the x-coordinates agree. This situation can be handled as a special case. Note that the fields k and K in this discussion are completely general. For example, k could be the field of p elements for some prime p, and K could be the field of q = p e elements for some positive integer e.

6 6 MATH 361: NUMBER THEORY FIRST LECTURE Now define 6. Rational Parametrization of the Circle L : x = 0, C : x 2 + y 2 = 1, and let P = ( 1, 0), an element of C k for any field k. Given a point Q = (x Q, y Q ) C K, the corresponding point on L K is ( ) y Q R = 0,. x Q + 1 Conversely, given a point R = (0, y R ) L K, let t = y R. We seek a point Q = (x, t(x + 1)) C K. But so we want x 2 + y 2 = ((x + 1) 1) 2 + t 2 (x + 1) 2 = (1 + t 2 )(x + 1) 2 2(x + 1) + 1, (1 + t 2 )(x + 1) 2 2(x + 1) = 0, or (1 + t 2 )(x + 1) = 2, or x + 1 = 2/(1 + t 2 ). Since y = t(x + 1) it follows that y = 2t/(1 + t 2 ), so that finally, ( ) 1 t 2 Q = 1 + t 2, 2t 1 + t An Application From Calculus Let θ denote the angle to a point (x, y) C R. Then the quantity t in the previous discussion is t = tan(θ/2). Thus θ = 2 arctan(t), giving the third of the equalities cos(θ) = 1 t2 2t 2 dt, sin(θ) =, dθ = 1 + t2 1 + t2 1 + t 2. The rational parametrization of the circle gives rise to the substitution in elementary calculus that reduces any integral of a rational function of the transcendental functions cos(θ) and sin(θ) of the variable of integration θ to the integral of a rational function of the variable of integration t, R(cos(θ), sin(θ)) dθ = R(t) dt. In the abstract, this last integral can be evaluated by the method of partial fractions, but doing so in practice requires factoring the denominator of the integrand. 8. Pythagorean Triples Again Again consider a primitive Pythagorean triple, (x, y, z) Z 3, x 2 + y 2 = z 2, x, y, z Z +, gcd(x, y, z) = 1,, x odd, y even. Let x = x/z and ỹ = y/z. Then ( x, ỹ) is a point of C Q, ( ) 1 t 2 ( x, ỹ) = 1 + t 2, 2t 1 + t 2, t = s/r Q.

7 MATH 361: NUMBER THEORY FIRST LECTURE 7 It follows that ( r 2 s 2 ) ( x, ỹ) = r 2 + s 2, 2rs r 2 + s 2, s, r Z. Here we take 0 < s < r, gcd(r, s) = 1. If in addition, r and s have opposite parities then the quotients will be in lowest terms, so that as before, x = r 2 s 2, y = 2rs, z = r 2 + s Cubic Curves Rather than a conic curve, we could consider a cubic curve, e.g., E : y 2 = x 3 g 2 x g 3 where g 2 and g 3 are constants. If the curve E has a rational point (x, y) then one of the magical phenomena of mathematics arises: the curve carries the structure of an abelian group. The subtleties of the group give rise to applications in connection with number theory (Fermat s Last Theorem in particular) and cryptography. Relevant references here are the book by Silverman and Tate, and the book by Washington.

### 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

### Geometry and Arithmetic

Geometry and Arithmetic Alex Tao 10 June 2008 1 Rational Points on Conics We begin by stating a few definitions: A rational number is a quotient of two integers and the whole set of rational numbers is

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

### Congruent Number Problem

University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases

### Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

### 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

### Solutions to Practice Problems

Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

### PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

MODULAR ARITHMETIC KEITH CONRAD. Introduction We will define the notion of congruent integers (with respect to a modulus) and develop some basic ideas of modular arithmetic. Applications of modular arithmetic

### Pythagorean Triples Pythagorean triple similar primitive

Pythagorean Triples One of the most far-reaching problems to appear in Diophantus Arithmetica was his Problem II-8: To divide a given square into two squares. Namely, find integers x, y, z, so that x 2

### Number Theory 2. Paul Yiu. Department of Mathematics Florida Atlantic University. Spring 2007 April 8, 2007

Number Theory Paul Yiu Department of Mathematics Florida Atlantic University Spring 007 April 8, 007 Contents 1 Preliminaries 101 1.1 Infinitude of prime numbers............... 101 1. Euclidean algorithm

### 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,

### 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

### The Equation x 2 + y 2 = z 2

The Equation x 2 + y 2 = z 2 The equation x 2 + y 2 = z 2 is associated with the Pythagorean theorem: In a right triangle the sum of the squares on the sides is equal to the square on the hypotenuse. We

### Topics in Number Theory

Chapter 8 Topics in Number Theory 8.1 The Greatest Common Divisor Preview Activity 1 (The Greatest Common Divisor) 1. Explain what it means to say that a nonzero integer m divides an integer n. Recall

### 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

### Pythagorean Triples and Fermat s Last Theorem

Pythagorean Triples and Fermat s Last Theorem A. DeCelles Notes for talk at the Science Speakers Series at Goshen College Document created: 05/10/011 Last updated: 06/05/011 (graphics added) It is an amazing

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### 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

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### Trigonometric Substitutions.

E Trigonometric Substitutions. E. The Area of a Circle. The area of a circle of radius (the unit circle) is well-known to be π. We will investigate this with several different approaches, each illuminating

### ARITHMETIC PROGRESSIONS OF THREE SQUARES

ARITHMETIC PROGRESSIONS OF THREE SQUARES KEITH CONRAD 1 Introduction Here are the first 10 perfect squares (ignoring 0): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 In this list is the arithmetic progression

### 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

### 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

### 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

### Right triangles and elliptic curves

Right triangles and elliptic curves Karl Rubin Ross Reunion July 2007 Karl Rubin (UCI) Right triangles and elliptic curves Ross Reunion, July 2007 1 / 34 Rational right triangles Question Given a positive

### Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms

The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### ARITHMETIC PROGRESSIONS OF FOUR SQUARES

ARITHMETIC PROGRESSIONS OF FOUR SQUARES KEITH CONRAD 1. Introduction Suppose a, b, c, and d are rational numbers such that a 2, b 2, c 2, and d 2 form an arithmetic progression: the differences b 2 a 2,

3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where

### PROBLEM SET # 2 SOLUTIONS

PROBLEM SET # 2 SOLUTIONS CHAPTER 2: GROUPS AND ARITHMETIC 2. Groups.. Let G be a group and e and e two identity elements. Show that e = e. (Hint: Consider e e and calculate it two ways.) Solution. Since

### Is this number prime? Berkeley Math Circle Kiran Kedlaya

Is this number prime? Berkeley Math Circle 2002 2003 Kiran Kedlaya Given a positive integer, how do you check whether it is prime (has itself and 1 as its only two positive divisors) or composite (not

### 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

New Zealand Mathematical Olympiad Committee Divisibility and Primes Arkadii Slinko 1 Introduction The theory of numbers is devoted to studying the set N = {1, 2, 3, 4, 5, 6,...} of positive integers, also

### 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

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### 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.

### Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

### 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

### MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS

MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:00-3:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:

### vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

### Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.

Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,

### The Three Famous Problems of Antiquity

The Three Famous Problems of Antiquity 1. Dupilication of the cube 2. Trisection of an angle 3. Quadrature of the circle These are all construction problems, to be done with what has come to be known as

### Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture. Brad Groff

Congruent Numbers, the Rank of Elliptic Curves and the Birch and Swinnerton-Dyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1

### 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.

### A Non-Existence Property of Pythagorean Triangles with a 3-D Application

A Non-Existence Property of Pythagorean Triangles with a 3-D Application Konstantine Zelator Department of Mathematics and Computer Science Rhode Island College 600 Mount Pleasant Avenue Providence, RI

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### 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

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS. 1. Ideals in Commutative Rings In this section all groups and rings will be commutative.

ALGEBRA HANDOUT 2: IDEALS AND QUOTIENTS PETE L. CLARK 1. Ideals in Commutative Rings In this section all groups and rings will be commutative. 1.1. Basic definitions and examples. Let R be a (commutative!)

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### An introduction to number theory and Diophantine equations

An introduction to number theory and Diophantine equations Lillian Pierce April 20, 2010 Lattice points and circles What is the area of a circle of radius r? You may have just thought without hesitation

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### Equivalence relations

Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence

### ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM

ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We give

### 4.2 Euclid s Classification of Pythagorean Triples

178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple

### Orders Modulo A Prime

Orders Modulo A Prime Evan Chen March 6, 2015 In this article I develop the notion of the order of an element modulo n, and use it to prove the famous n 2 + 1 lemma as well as a generalization to arbitrary

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Fibonacci Numbers Modulo p

Fibonacci Numbers Modulo p Brian Lawrence April 14, 2014 Abstract The Fibonacci numbers are ubiquitous in nature and a basic object of study in number theory. A fundamental question about the Fibonacci

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Introduction Proof by unique factorization in Z Proof with Gaussian integers Proof by geometry Applications. Pythagorean Triples

Pythagorean Triples Keith Conrad University of Connecticut August 4, 008 Introduction We seek positive integers a, b, and c such that a + b = c. Plimpton 3 Babylonian table of Pythagorean triples (1800

### EULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.

EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If

### 4 Number Theory I: Prime Numbers

4 Number Theory I: Prime Numbers Number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. Despite their ubiquity and apparent simplicity, the natural

### MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

### 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

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### Chapter Two. Number Theory

Chapter Two Number Theory 2.1 INTRODUCTION Number theory is that area of mathematics dealing with the properties of the integers under the ordinary operations of addition, subtraction, multiplication and

### TRIANGLES ON THE LATTICE OF INTEGERS. Department of Mathematics Rowan University Glassboro, NJ Andrew Roibal and Abdulkadir Hassen

TRIANGLES ON THE LATTICE OF INTEGERS Andrew Roibal and Abdulkadir Hassen Department of Mathematics Rowan University Glassboro, NJ 08028 I. Introduction In this article we will be studying triangles whose

### Total Gadha s Complete Book of NUMBER SYSTEM

Total Gadha s Complete Book of NUMBER SYSTEM TYPES OF NUMBERS Natural Numbers The group of numbers starting from 1 and including 1, 2, 3, 4, 5, and so on, are known as natural numbers. Zero, negative numbers,

### PERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES. 1. Introduction

PERIMETERS OF PRIMITIVE PYTHAGOREAN TRIANGLES LINDSEY WITCOSKY ADVISOR: DR. RUSS GORDON Abstract. This paper examines two methods of determining whether a positive integer can correspond to the semiperimeter

### 9 abcd = dcba. 9 ( b + 10c) = c + 10b b + 90c = c + 10b b = 10c.

In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should

### Theorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive

Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime

### Montana Common Core Standard

Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in

### MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS

MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### Characterizing the Sum of Two Cubes

1 3 47 6 3 11 Journal of Integer Sequences, Vol. 6 (003), Article 03.4.6 Characterizing the Sum of Two Cubes Kevin A. Broughan University of Waikato Hamilton 001 New Zealand kab@waikato.ac.nz Abstract

Albuquerque School of Excellence Math Curriculum Overview Grade 11- Algebra II Module Polynomial, Rational, and Radical Relationships( Module Trigonometric Functions Module Functions Module Inferences

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### Continued Fractions. Darren C. Collins

Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

### Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto September 5, 2005 Numbers and Equations

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### Definition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.

Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### 3 Congruence arithmetic

3 Congruence arithmetic 3.1 Congruence mod n As we said before, one of the most basic tasks in number theory is to factor a number a. How do we do this? We start with smaller numbers and see if they divide

### Chapter 1 Basic Number Concepts

Draft of September 2014 Chapter 1 Basic Number Concepts 1.1. Introduction No problems in this section. 1.2. Factors and Multiples 1. Determine whether the following numbers are divisible by 3, 9, and 11:

### MAT Mathematical Concepts and Applications

MAT.1180 - Mathematical Concepts and Applications Chapter (Aug, 7) Number Theory: Prime and Composite Numbers. The set of Natural numbers, aka, Counting numbers, denoted by N, is N = {1,,, 4,, 6,...} If

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

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

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

### Our Primitive Roots. Chris Lyons

Our Primitive Roots Chris Lyons Abstract When n is not divisible by 2 or 5, the decimal expansion of the number /n is an infinite repetition of some finite sequence of r digits. For instance, when n =

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### 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

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod