# Our Primitive Roots. Chris Lyons

Size: px
Start display at page:

## Transcription

1 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 = 7 we know that /7 = which repeats the 6-digit sequence Given n, how long can r possibly be? We ll start by exploring this simple question, which will ultimately lead us to the notion of primitive roots and to a long standing unsolved problem. Then we ll narrow our focus to the expansion of /p when p is prime. In the case when p is of the form 4k + 3 and the repeating sequence is as long as possible, it turns out that these seemingly mundane digits actually encode a deep property of the prime number p. Decimal expansions of rational numbers At some point in your mathematical education, you ve learned the following distinction between rational and irrational real numbers: roughly speaking, the decimal expansions of rational numbers are always either finite in length or eventually periodic, while the decimal expansions of irrational numbers never terminate and are never periodic. I want to focus on the decimal expansions of rational numbers. These are, in a mathematical sense, everyday objects. Indeed, whether you chose to or not, I m sure you ve memorized at least a few of them. For instance, you probably know /2 = 0.5 /3 = /4 = 0.25 /5 = 0.2 /6 = How about the expansion of /7? You ve certainly seen it: /7 = but perhaps the repeating part, which is the 6 digit sequence 42857, is a little too long to stick in your memory. Similarly, you may know the expansion: / = but if you boost the denominator up by 2 then you probably haven t memorized /3 = which repeats the 6 digit sequence Of course we could talk about fractions that look different than /n, e.g., 7/3, 9/6, etc. But any fraction is just a multiple of /n for some n, so we ll stick to /n for our discussion. Much (though not all) of what we ll say is relatively simple to extend to more general fractions.

2 Before proceeding further, let s remember a handy notational device that you ve also probably seen before: if a sequence of digits repeats, we just put a line over that sequence to indicate this. Three examples are: /3 = 0.3, /6 = 0.6, /7 = When does the decimal expansion of /n terminate, and when does it go on forever? If it does go on forever, when does it start out with some initial non-repeating part before reaching a repeating sequence? The answer is summarized in the following: Proposition.. Let n 2 be an integer. (a) The decimal expansion of /n terminates if and only if n is not divisible any primes other than 2 and 5. (b) Suppose the decimal expansion of /n does not terminate (meaning n is divisible by at least one prime not equal to 2 or 5). Then the decimal expansion is of the form /n = 0.a a 2... a r for some digits a,..., a r if and only if n is divisible by neither 2 nor 5. As an example of (b), notice that the decimal expansions /6 = 0. 6 and /2 = both begin with a short non-repeating sequence, and this can be explained by the fact that both 6 and 2 are divisible by 2. Proof. If you re used to writing proofs, then you can probably come up with one after awhile by playing around with some examples to see what s going on. (For instance, you might notice that and that 80 = = = = = ( 8 + ) = , 00 3 and think about some other similar examples; why have I written them in this way?) Once you have the idea, it becomes a matter of carefully setting up the right notation to cover the general case. Since this gets a little messy and will lead us off on a sidetrack, we ll omit the proof. We re only going to concern ourselves with the case when the decimal expansion is infinite and has no non-repeating part. Thus: From now on we assume that n 2 is divisible by neither 2 nor 5. Equivalently, we assume that gcd(n, 0) = ; often we describe this situation by saying that n is coprime to 0. So we re assuming that the decimal expansion is just an infinite repetition of some finite sequence. There s a little bit of ambiguity here. For instance, all three of the expressions /3 = 0. 3 = 0.33 = are accurate, but the first is the most efficient. Here s how we handle this ambiguity: Definition.2. We ll say that the period of /n is the length of the smallest repeating sequence in the decimal expansion of /n. Hence the period of /3 is and the period of /7 is 6. (I should warn you that this terminology may not be standard, but it will be convenient for us.) 2

3 Now let s explore this notion of period by looking at a list of decimal expansions of /n for the first few values of n that are not divisible by 2 or 5: /3 = 0. 3 /7 = /9 = 0. / = 0.09 /3 = /7 = /9 = /2 = /23 = /27 = /29 = /3 = Let s start looking for patterns, even some vague ones. For starters, we notice that the periods seem to be getting longer as n increases. However, at n = 2 and n = 27, the period becomes short again. Why? Well, a good guess is that it has something to do with the fact that 2 and 27, although of comparable size with numbers like 9, 23, and 29, are both divisible by smaller numbers (3 and 7). So a rough first guess could be: If n is divisible only by small primes, then the period of the expansion of /n will be relatively small. This would seem to imply in particular that if n is a large prime number then the period of /n should be large. This is what s happening for n = 7, 9, 23, 29, and 3. But if we go a little higher, this philosophy runs into trouble: /37 = 0.037, /4 = , /73 = Compared to the size of n, these expansions have pretty small periods! If you were to make a table of the periods of /n as n grows, even if you just restricted to prime values of n, you would find that patterns are hard to come by. 2 Primitive roots To say anything further, we need to think about how we go from infinite repeating decimal expansions back to fractions. Let s write /n = 0.a a 2... a r where each a i is a decimal digit between 0 and 9 and r is the period of /n. Let s have m be the integer which, in decimal notation, is written as a a 2... a r. (Example: when n = 3 we have /3 = so in this case m = = ) Then what the decimal expansion actually means is n = m 0 r + m 0 2r + m 0 3r +... = m 0 jr. We use the geometric series formula to get the right side into a nice closed form: n = m 0 jr = m ( ) j 0 r = m/0r /0 r = m 0 r. j= j= Put another way, we have 0 r = mn, which means that n divides 0 r. In fact, we have the following stronger statement: j= 3

4 Proposition 2.. The period of /n is the smallest integer r such that n divides 0 r. Proof. We ve already shown that if the period of /n is r, then n divides 0 r. What remains is to show the following: if s < r, then 0 s is not divisible by n. Assume for contradiction that we have 0 s = nq. Then we can just reverse the work we did above: we have n = q 0 s = j= q 0 jr. Notice that if the decimal representation of q is d d 2... d l, then we have l s (because q divides 0 s, which is the s-digit number ). So the sum on the right represents a repeating decimal expansion of period s. Since s < r, and r was defined to be the period of /n, this is a contradiction. As an example of this, consider n = 7. We have 0 = 9 = 3 2 ; 0 2 = 99 = 3 2 ; 0 3 = 999 = ; 0 4 = 9999 = 3 2 0; 0 5 = = ; 0 6 = = Since r = 6 is the smallest positive integer such that 0 r is divisible by 7, it follows that the period of /7 is equal to 6 (which we ve already seen before). Believe it or not, it s actually advantageous to phrase Proposition 2. in more abstract language. If you re familiar with the language of congruences, then the proposition above is equivalent to saying the following: Proposition 2.2. The period of /n is the smallest integer r such that 0 r (mod n). But the abstraction doesn t stop there. We can in turn rephrase this using basic notions from abstract algebra. Don t worry if you don t have background in this, because we ll just be briefly passing through this world. You can just gloss over the unknown technical terms and pay attention to the concrete consequences in Proposition 2.4 below... and take it as an advertisement for the power of abstract algebra! In abstract algebra you learn, for every integer m, about the ring of congruence classes modulo m, which is denoted as Z/mZ. (Sometimes this ring is also denoted as Z m, although number theorists tend to furrow their brows at this notation... What makes them so fussy? The keyword is p-adic integers. ) Inside of this ring is the group of multiplicative units, denoted (Z/mZ). Since we re assuming that n is coprime to 0, this means that the congruence class [0] Z/nZ is actually contained in the group (Z/nZ). Using this terminology, Proposition 2. can be rephrased as: Proposition 2.3. The period of /n equals the order of the congruence class [0] in the group (Z/nZ). This formulation, while perhaps unpleasantly abstract at first, is powerful because it leads more directly to the following concrete consequences: Proposition 2.4. Suppose that n = p k pk pk d d (a) The period of /n must divide is the prime factorization of n, with each k i. Then p k (p )p k2 2 (p 2 )... p k d d (p d ). (b) Let the period of /p ki be r i. Then the period of /n is equal to the least common multiple lcm(r, r 2,..., r d ). 4

5 Proof. One can show that the size of the group (Z/nZ) is ϕ(n) = p k (p )p k2 (p 2 )... p k d (p d ), where ϕ is the Euler phi function. By the basic theory of finite groups (specifically, Lagrange s Theorem), the order of any element in (Z/nZ) divides the order of the group, so we get part (a). Part (b) follows from a result in number theory or ring theory (depending upon your taste) called the Chinese Remainder Theorem. This result says and then yields (Z/nZ) (Z/p k Z) (Z/pk2 2 Z)... (Z/pk d d Z), (Z/nZ) (Z/p k Z) (Z/p k2 2 Z)... (Z/p k d d Z). This second isomorphism is what is needed to show (b). This proposition is quite an eyeful, but if you step back from it, you ll notice that it says that the prime factorization of n is very important in determining the period of /n. In fact it says that we ll be able to understand the period of /n, for any n, if we re able to understand the period of /p k for primes p and k. Instead of considering the larger case when n is a general power of a prime number, we ll just focus our attention on the first rung of this ladder by letting n be prime. From now on, we assume that p 2, 5 is a prime. In this case Proposition 2.4 says the following: Corollary 2.5. The period of /p divides p. So for instance, I may not know off the top of my head what the period of /997 is, but I do know that it has to divide 996 = Thus the period belongs to the (relatively small!) set of possibilities {, 2, 3, 4, 6, 2, 83, 66, 249, 332, 498, 996}. Let s explore the periods of /p for the some small primes. We ve just said that the period of /p divides the integer p. Let p k denote the kth prime number (so p = 2, p 2 = 3, p 3 = 5,...). In the chart below, I ve plotted the points ( k, period of /p ) k p k for k 00. (Notice the anomalous values at p = 2 and p 3 = 5; we re not supposed to consider these, so I ve just set the value to 0.) 5

7 The length r of the shortest repeating sequence in the base b expansion of /n is the smallest integer r such that b r is divisible by n. When p is prime, this length r divides p, and we have r = p exactly when b is a primitive root of p. When b is not a square, then Emil Artin generalized Gauss conjecture above by postulating that that there are infinitely many primes p such that b is a primitive root of p. 4 How random-looking are the digits of /p? Now we re going to return to decimal expansions and narrow our focus: From now on p will be a prime for which the period of /p equals p. When p is large, this means the repeating sequence of p digits is pretty long. Let s look at p = 47, which has a period of 46: /47 = The first few digits will be fairly predictable, because we know /47 /50 = But past a certain point the digits start looking, well, kind of random. Of course the digits of the decimal expansion of /p aren t actually random at all, but what I mean is this: If we take out a string of digits from the middle of the decimal expansion of /p and compare it with actual random strings of digits, can we tell the difference? Let s try it. In the five rows below are strings of 35 digits. One of the rows is a string of 35 digits taken from the middle of the expansion of /3 (which as period 30), one of them is from the middle of the expansion of /227 (which has period 226), and the other three rows are strings of 35 random digits. Which of these three rows are random strings? It s pretty tough to tell with the naked eye. The answer is that the first row comes from /3 and the last row comes from /227. So now that we ve convinced ourselves that these strings look somewhat random to our eyes, let s go beyond this and use a statistical test. There tons of tests that we could apply here, but I want to focus on one particular test for randomness. As we ll see, this test is going to fail when applied to the digits of /p, and it will fail in a spectacular way. Here s the test: if c c 2... c p is any string of p digits, we ll say that the A-statistic of this string is the alternating sum p A := ( ) j c j = c + c 2 c 3 + c 4... c p 2 + c p. j= Now if c c 2... c p is a truly random string of digits (to be more precise for those who know probability: assuming the digits are chosen with uniform probability from 0 through 9, and chosen independently) and p is large then: (a) The value of A is likely to be close to zero (since p is even and the expected value of all digits is the same, namely 4.5), and yet it s unlikely to actually equal zero. (b) The value of A is just as likely to be positive as it is to be negative. 7

8 Knowing what ought to happen for random strings of p digits, let s now apply the A-statistic to the digits of /p. In the following table, the numbers in the left column are the first few primes p for which /p has period p. If /p = 0.a a 2... a p, then the right column is the A-statistic of the string a a 2... a p. prime p for which period A-statistic applied to of /p equals p repeating digits of /p Let s evaluate the randomness of the digits of /p based upon the A-statistic. First off, in half of the cases, A is exactly equal to zero, which we said above in (a) is unlikely to happen in the random case. Secondly, in the cases when it s nonzero, A is never actually negative, which conflicts with our criteria (b) for randomness. So, if this sort of behavior continues for larger and larger primes, then these expansions don t look very random at all through the lens of the A-statistic! This is already interesting, but let s look a little more closely at the positive numbers in the right column: they re all multiples of. This peculiar behavior of the digits of /p was observed only within the last 20 years, by number theorist Kurt Girstmair: Theorem 4. (Girstmair 994). Suppose that p is a prime number such that the period of /p is p. If /p = 0.a a 2... a p, then the A-statistic of the string a a 2... a p is { 0 if p = 4k + for some k A = h p if p = 4k + 3 for some k, where h p is a certain positive integer depending upon p. A looming question is: when p = 4k + 3, what is this integer h p? 5 Class numbers To answer that question, we have make a foray into the world of algebraic number theory. If p = 4k + 3 for some k, let s consider the following set of complex numbers: { ( ) + i p } R p := a + b a, b Z C. 2 This set is closed under addition, subtraction, and multiplication and is, in the parlance of abstract algebra, a ring. The world s most famous ring is Z, the ring of integers. Number theorists were originally interested 8

10 6 Further Reading The most accessible account of Girstmair s result (which also contains an alternative discussion of the class number h p ) is: K. Girstmair. A popular class number formula. Amer. Math. Monthly, Vol. 0, No. 0 (Dec., 994), pp The result there, which is the one discussed in this exposition, is a special case of more technical work here: K. Girstmair. The digits of /p in connection with class number factors. Acta Arith., Vol. 67, No. 4 (994), pp In a different direction, Murty and Thangadurai have studied the digits of the (decimal or other base) expansion of /p when the period is not necessarily p : M. Ram Murty and R. Thangadurai. The class number of Q( p) and the digits of /p. Proc. Amer. Math. Soc., Vol. 39, No. 4 (20), pp

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### 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.

### 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

### 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

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### 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

### 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

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### 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.

### 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

### 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

### 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

### 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

### 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

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### 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

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### 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

### 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

### 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

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### 47 Numerator Denominator

JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### 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(

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

### 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

### 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

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

### 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)

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### 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

### THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

### 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works.

MATH 13150: Freshman Seminar Unit 18 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. 1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the internet

### 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.

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b

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

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### 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

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Ummmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:

Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### 23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

### I remember that when I

8. Airthmetic and Geometric Sequences 45 8. ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I

### 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

### RSA Encryption. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003

RSA Encryption Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003 1 Public Key Cryptography One of the biggest problems in cryptography is the distribution of keys.

### 3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 1 Real Numbers

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 1 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

### 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

### 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)

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 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

### JobTestPrep's Numeracy Review Decimals & Percentages

JobTestPrep's Numeracy Review Decimals & Percentages 1 Table of contents What is decimal? 3 Converting fractions to decimals 4 Converting decimals to fractions 6 Percentages 6 Adding and subtracting decimals

### 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)

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography

V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### 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:

### 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

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### 1. The Fly In The Ointment

Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.

[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.

31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The

### MATH 13150: Freshman Seminar Unit 10

MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers

### 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

### 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 +