Math 5330 Spring Notes Prime Numbers

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 5330 Spring Notes Prime Numbers"

Transcription

1 Math 5330 Sring 206 Notes Prime Numbers The study of rime numbers is as old as mathematics itself. This set of notes has a bunch of facts about rimes, or related to rimes. Much of this stuff is old dating back years. We start with ossibly the most imortant question about rimes: how many are there? We have an ancient method for calculating lists of rimes, which is still unbelievable good. It is called the Sieve of Erotosthenes. The idea is as follows: Form an array of numbers, 2, 3,.... Ignore because it is a unit. Circle 2, and cross out every subsequent multile of 2. Suose that the last number in your array is n. Reeat the following until you circle a number greater than? n : Find the next number not crossed off, circle it, and cross off every subsequent multile. Once a number greater than? n is circled, dont cross off any additional numbers, just circle the remaining numbers not crossed off, they will all be rime. For examle, to find all rimes less than 00 takes 4 stes:

2 This algorithm is very fast. Ive heard it said that if you are writing a rogram that makes use of all rimes less than,000,000 it is faster to use the sieve of Erotosthenes than to read in a reexisting file of rimes. One thing that makes the algorithm so fast is that we can sto sieving once we reach? n. The reason for this is the following theorem. Theorem If n is not rime, then n has a rime divisor ď? n. Proof: Let n be comosite and let q be a rime divisor of n, so n qn{qq. If q ď? n, then let q, in the theorem. Otherwise, n{q ă? n, so let be any rime divisor of n{q. As a consequence of this theorem, we have the following factoring technique: To factor a number n, try dividing n in turn by the rimes 2, 3, 5,.... We continue until a rime gets larger than the square root of the unfactored art. For examle, suose we wish to factor We divide by 2 until an odd number results: The unfactored art is not divisible by 3, but it is by 5, so we have Continuing, we have Note that after dividing by 3 to leave an unfactored art of 0, we can sto because 0 ă 3 2. This means that 0 must be rime. At this oint, we have a good way to find all small rimes, and a reasonable algorithm for factoring small numbers. We still have not answered the question of how many rimes there are. Consider following table: range ` ` rime count This table gives the number of rimes in ranges of 00 consecutive integers. Based on the table, one might exect that the number of rimes is finite and that there is some largest rime. However, this is not the case. The following is a result due to Euclid: Theorem 2 There are infinitely many rimes. Proof: Suose not, and let t2, 3,..., k u be a comlete list of rimes. Let M be the number M k `. (M is one greater than the roduct of all rimes.) When M is divided by 2 or 3 or... or k, the remainder will be. Thus, M is not divisible by any of the rimes Page 2

3 in our list. But the Fundamental Theorem says M is divisible by some rime. This contradicts the assumtion that our list was comlete, which comletes the roof of the theorem. Given the first k rimes, we can define the number M k 2 3 k `. The first several are M 3, M 2 7, M 3 3, M 4 2. These first ones are rimes and it is a common conjecture by students that M k is always rime. But this is not the case: M It turns out that rimes are fairly common. We give a name to the number of rimes u to some bound: πnq the number of rimes ď n. For examle, π00q 25 because there are 25 rimes less than 00. In 793 at the age of 5, Gauss made the following conjecture about how common rime numbers are: πxq «ż x 2 lntq dt. This integral has a name, it is called lixq, the logarithmic integral. In 896, this conjecture was roved roughly simultaneously by Hadamard and de la Vallée Poussin. The exact statement of the result is this: Theorem 3 (The Prime Number Theorem or PNT) πxq lim xñ8 lixq. It turns out that lixq «x, so we usually say πxq «x lnxq htt://mathworld.wolfram.com/primenumbertheorem.html. Here is a grah from lnxq Page 3

4 The following table also gives a feel for how good these aroximations are. n n πnq lnnq linq ,000,000 78,498 72,382 78,626,000,000,000 50,847,478 48,254,942 50,849,234 It took about 00 years to rove the Prime Number Theorem because it took that long to develo the necessary results in Comlex Analysis. What does the field of analysis have to do with rime numbers? To give a feel, here is a second roof that there are infinitely many rimes, making use of analysis. This roof will also show how logarithms might enter into the icture, and how many different areas of mathematics seem to be related to number theory. Our actual theorem is the following: The sum ÿ 2 ` 3 ` 5 ` 7 ` ` diverges. This shows there are infinitely many rimes because any finite sum would converge. Moreover, it shows that rimes must be fairly common, much more common than squares, 8ÿ because the sum of the recirocals of the squares, n converges. In fact, ÿ 8 2 n π n n This isn t relevant, just an interesting fact. Our roof is indirect, and is essentially due to Euler. What he did was to first consider the roduct ź ` ` ` 2 `. 3 ďn For examle, when n 0, the roduct is ` 2 ` 2 ` ` 2 3 ` 3 ` ` 2 5 ` 5 ` ` 2 7 ` 7 `. 2 If we were to multily this out, the sum would be ` 2 ` 3 ` 4 ` 5 ` 6 ` 7 ` 8 ` 9 ` 0 ` 2 ` 4 ` 5 ` 6 ` 8 `. The denominators are those divisible by 2 s, 3 s, 5 s, and 7 s. For examle, to get 8 we multily the terms 2 from the first roduct, from the second, and the s from the third 32 and fourth roducts. In articular, ź ďn ` ` ` 2 ` ą 3 nÿ k k ą ln n. Page 4

5 The reason nÿ k k ż ą ln n is that the sum can be aroximated by the integral dx. In x ż n dx, we get the x 2 ` 3 ` 4 ` 5 ` 6 ` 7 ` ż 9 8 ą dx ln9q. That is, the x articular, if we use uer rectangles to estimate the area of lnnq sum. For examle, below, ` sum is actually larger than lnn ` q, but lnnq looks a little simler. Now the sum is a geometric series with sum have ź ` ` ` 2 ` 3 ďn. Putting the ieces together so far, we ą ln n. Next, we take the logarithm of this to convert the roduct to a sum giving ÿ ln ą ln ln n. ďn To get to the sum of the recirocals of the rimes, we could use the MacLauran exansion: ln xq x ` x2 2 ` x3 3 ` but it works out better to use the formula ln xq ď x ` x2 x 2, for 0 ď x ă. A grah will show this is true. Alternatively, if you let fxq x ` x2 ` ln xq, x2 Page 5

6 then f0q 0 and f xq ą 0 for all 0 ă x ă, so fxq starts at 0 and increases. We have ln ln n ă ÿ ln ď ÿ f ÿ ` ÿ 2. ďn ďn ďn ďn One last trick: ÿ ďn 8 2 ď ÿ k 2 k ÿ It isn t imortant that k 2 3, only that it is some finite number, but it is hard to 4 k 2 ass u a nice formula. To see this summation is correct, we use artial fractions: k 2 2 k, k ` so 8ÿ 8 k 2 ÿ 2 k k ` k ` 2 4 ` 3 5 ` 4 6 ` 5 ` ` k 2 Putting everything together, n ln n ă ÿ ďn ` 3 4 or ÿ ďn ą ln ln n 3 4. Since ln ln n Ñ 8 as n Ñ 8, the sum of the recirocals of the rimes diverges. Perfect numbers A number is called a erfect number if it equals the sum of its roer divisors (or if the sum of all the divisors is twice the number.) The first several erfect numbers are 6 ` 2 ` 3 28 ` 2 ` 4 ` 7 ` ` 2 ` 4 ` 8 ` 6 ` 3 ` 62 ` 24 ` ` 2 ` 4 ` 8 ` 6 ` 32 ` 64 ` 27 ` 254 ` 508 ` 06 ` 2032 ` Looking at the rime factorizations of these numbers, we have 6 2 3, , , This might lead us to guess that erfect numbers always have the Page 6

7 form 2 k for some rime number. We can say more: If n 2 k, then the factors of n are, 2, 2 2,..., 2 k,, 2, 2 2,..., 2 k. These consist of two geometric sequences, and their sum is 2 k` q ` 2 k` q. On the other hand, we want the sum to be 2n 2 k`. For this to be the case, we need 2 k` q ` 2 k` q 2 k` Ñ 2 k` 0, or 2 k`. This means we can t use any k and, they must be linked. In articular, we can only use rimes that are one less than a ower of 2. The mathematician Mersenne looked into this in some deth. Let Mnq 2 n. We call such numbers Mersenne numbers. If Mnq is rime, we refer to it as a Mersenne rime. We have roven the following. Theorem 4 If 2 n is rime, then m 2 n is a erfect number. Do all erfect numbers have the form 2 n where 2 n? We don t know. Here is a artial answer, however. Theorem 5 (Euler) If m is a erfect number and m is even, then m 2 n 2 n q where 2 n is rime. Proof: Let m be an even erfect number, and suose that m 2 k Q where Q is odd. Suose the sum of the divisors of Q is S. Then the sum of the divisors of m is S ` 2 ` 2 2 ` ` 2 k q S2 k` q. You should convince yourself that the sum, indeed, looks like this. It is not quite obvious. As above, we want the sum to be 2m so 2m S2 k` q. Solving for S, S 2m 2k` 2 k` Q 2 k` Q ` Q 2 k`. Q But S is the sum of all the divisor of Q and Q and are both divisors of Q. This 2 k` means that Q cannot have any other divisors and the only way this could haen is if Q is Q rime, and 2 k` giving Q 2k`. This comletes the roof. Some obvious questions: Question : Are there any odd erfect numbers? Question 2: Are there infinitely many erfect numbers? No one knows the answers to these questions, though there is strong circumstantial evidence that there are infinitely many erfect numbers. The second question could be osed in terms of Mersenne rimes: are there infinitely many Mersenne rimes? Again, it is thought that the answer is yes. What does it take for a Mersenne number to be a Mersenne rime? Here is a table of the first several Mersenne numbers. n n Page 7

8 A attern resents itself: only rime numbers n can give rise to rimes (in articular, ) In fact, Mersenne numbers have a multilicative roerty: If m n then Mmq Mnq. For examle, 7 M3q divides M9q. This is not too hard to rove: If n km then 2 n 2 km 2 m q k 2 m q2 mk q ` 2 mk 2q ` ` 2 m ` q, so 2 n has 2 m as a factor. Unfortunately, even for rimes,, 2 need not be rime. For examle, As of February 203 there are 48 known rimes for which 2 is rime. these are: 2, 3, 5, 7, 3 the ones known to the Greeks, 7, 9, 3, 6 discovered before the 20th century, 89, 07, 27, 52, 607, 279, 2203, 228, 327 discovered before 960, 4253, 4423, 9689, 994, 23, 9937, 270, 23209, discovered before 980, 86243, 0503, 32049, 2609 found in the 980 s, , , , , , , found in the 990 s, 3,466,97 (200), 20,996,0 (2003), 24,036,583 (2004), 25,964,95, 30,402,457 (2005), 32,582,657 (2006), 43,2,609, 37,56,667 found in August, Setember, 2008, 42,643,80 found in Aril, 2009, 57,885,6 found January 25, 203, 74,207,28 found in January, 206. It is known that 2 32,582,657 is the 44 th Merseene rime. Also, all exonents u to 63,02,93 have been checked at least once so it is highly likely that 2 57,885,6 is the 48 th Mersenne rime. However, it is ossible that there are other Mersenne rimes left to discover between 57, 885, 6 and 74, 207, 28. Page 8

9 The following is an algorithm to check if a number 2 is a Mersenne rime. It is called the Lucas-Lehmer test. Set U 4. for i from 3 to do relace U by U 2 2 mod 2 q at the end of the loo, if U 0, then 2 is rime, otherwise, 2 is comosite. For examle, if 9, we have U 4 Ñ 4 Ñ 94 Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ 0, so 2 9 is a Mersenne rime. However, when 23, we have U 4 Ñ 4 Ñ 94 Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ so 2 23 is not a Mersenne rime. Page 9

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has

A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency

More information

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.

More information

As we have seen, there is a close connection between Legendre symbols of the form

As we have seen, there is a close connection between Legendre symbols of the form Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In

More information

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone

Algorithms for Constructing Zero-Divisor Graphs of Commutative Rings Joan Krone Algorithms for Constructing Zero-Divisor Grahs of Commutative Rings Joan Krone Abstract The idea of associating a grah with the zero-divisors of a commutative ring was introduced in [3], where the author

More information

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

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11) Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint

More information

TRANSCENDENTAL NUMBERS

TRANSCENDENTAL NUMBERS TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating

More information

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

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

More information

Chapter 11 Number Theory

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

More information

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 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

More information

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES

More information

Number Theory Naoki Sato <ensato@hotmail.com>

Number Theory Naoki Sato <ensato@hotmail.com> Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an

More information

Doug Ravenel. October 15, 2008

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

More information

Pythagorean Triples and Rational Points on the Unit Circle

Pythagorean Triples and Rational Points on the Unit Circle Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns

More information

Math Workshop October 2010 Fractions and Repeating Decimals

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,

More information

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006

The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 joshua.zucker@stanfordalumni.org [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theory

More information

Fractions and Decimals

Fractions and Decimals Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

More information

Lecture 21 and 22: The Prime Number Theorem

Lecture 21 and 22: The Prime Number Theorem Lecture and : The Prime Number Theorem (New lecture, not in Tet) The location of rime numbers is a central question in number theory. Around 88, Legendre offered eerimental evidence that the number π()

More information

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking

More information

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

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

More information

Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions

Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going

More information

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields Tom Weston Contents Introduction 4 Chater 1. Comlex lattices and infinite sums of Legendre symbols 5 1. Comlex lattices 5

More information

SECTION 10-2 Mathematical Induction

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

More information

8 Primes and Modular Arithmetic

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.

More information

Lecture 13 - Basic Number Theory.

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

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Number Theory Naoki Sato <sato@artofproblemsolving.com>

Number Theory Naoki Sato <sato@artofproblemsolving.com> Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material

More information

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

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

More information

1.2. Successive Differences

1.2. Successive Differences 1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers

More information

More Properties of Limits: Order of Operations

More Properties of Limits: Order of Operations math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f

More information

Complex Conjugation and Polynomial Factorization

Complex Conjugation and Polynomial Factorization Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.

More information

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

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

More information

Session 6 Number Theory

Session 6 Number Theory Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

More information

A Study on the Necessary Conditions for Odd Perfect Numbers

A Study on the Necessary Conditions for Odd Perfect Numbers A Study on the Necessary Conditions for Odd Perfect Numbers Ben Stevens U63750064 Abstract A collection of all of the known necessary conditions for an odd perfect number to exist, along with brief descriptions

More information

Integers and division

Integers and division CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

More information

Homework until Test #2

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

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

Elementary factoring algorithms

Elementary factoring algorithms Math 5330 Spring 013 Elementary factoring algorithms The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat s Little Theorem and some related ideas, one can

More information

Introduction to NP-Completeness Written and copyright c by Jie Wang 1

Introduction to NP-Completeness Written and copyright c by Jie Wang 1 91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of

More information

The Magnus-Derek Game

The Magnus-Derek Game The Magnus-Derek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.

More information

15 Prime and Composite Numbers

15 Prime and Composite Numbers 15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such

More information

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+ The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us

More information

Elementary Number Theory: Primes, Congruences, and Secrets

Elementary Number Theory: Primes, Congruences, and Secrets This is age i Printer: Oaque this Elementary Number Theory: Primes, Congruences, and Secrets William Stein November 16, 2011 To my wife Clarita Lefthand v vi Contents This is age vii Printer: Oaque this

More information

Continued Fractions and the Euclidean Algorithm

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

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

More information

Section 6-2 Mathematical Induction

Section 6-2 Mathematical Induction 6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

More information

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

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,

More information

THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE

THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE / 24 THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE Todd Cochrane Everything is Golden 2 / 24 Golden Ratio Golden Proportion Golden Relation Golden Rectangle Golden Spiral Golden Angle Geometric Growth, (Exponential

More information

Precalculus Prerequisites a.k.a. Chapter 0. August 16, 2013

Precalculus Prerequisites a.k.a. Chapter 0. August 16, 2013 Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College August 6, 0 Table of Contents 0 Prerequisites 0. Basic Set

More information

10.2 Series and Convergence

10.2 Series and Convergence 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

More information

A Modified Measure of Covert Network Performance

A Modified Measure of Covert Network Performance A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES lynnedoty@maristedu Abstract: In a covert network the need for secrecy

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

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

More information

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 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

More information

MATH 361: NUMBER THEORY FIRST LECTURE

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,

More information

An Introduction to Number Theory Prime Numbers and Their Applications.

An Introduction to Number Theory Prime Numbers and Their Applications. East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Homework 5 Solutions

Homework 5 Solutions Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

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

More information

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. 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

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

Prime Factorization 0.1. Overcoming Math Anxiety

Prime Factorization 0.1. Overcoming Math Anxiety 0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF

More information

5-4 Prime and Composite Numbers

5-4 Prime and Composite Numbers 5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee

More information

MATH 13150: Freshman Seminar Unit 10

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

More information

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method Universiteit-Utrecht * Deartment of Mathematics Otimal a riori error bounds for the Rayleigh-Ritz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL

More information

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Develoment FRANCE Synosys There is no doubt left about the benefit of electrication and subsequently

More information

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

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

More information

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

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

More information

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians = Introduction to Modeling 3.6-1 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants

More information

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID International Journal of Comuter Science & Information Technology (IJCSIT) Vol 6, No 4, August 014 SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

More information

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

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

More information

1 Gambler s Ruin Problem

1 Gambler s Ruin Problem Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins

More information

Stat 134 Fall 2011: Gambler s ruin

Stat 134 Fall 2011: Gambler s ruin Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some

More information

Today s Topics. Primes & Greatest Common Divisors

Today s Topics. Primes & Greatest Common Divisors Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime

More information

Basic Proof Techniques

Basic Proof Techniques Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

More information

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

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

More information

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

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

More information

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

(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

More information

Introduction. Appendix D Mathematical Induction D1

Introduction. Appendix D Mathematical Induction D1 Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

More information

k, then n = p2α 1 1 pα k

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

More information

Two classic theorems from number theory: The Prime Number Theorem and Dirichlet s Theorem

Two classic theorems from number theory: The Prime Number Theorem and Dirichlet s Theorem Two classic theorems from number theory: The Prime Number Theorem and Dirichlet s Theorem Senior Exercise in Mathematics Lee Kennard 5 November, 2006 Contents 0 Notes and Notation 3 Introduction 4 2 Primes

More information

Session 7 Fractions and Decimals

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,

More information

Assignment 9; Due Friday, March 17

Assignment 9; Due Friday, March 17 Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that

More information

DIRICHLET PRIME NUMBER THEOREM

DIRICHLET PRIME NUMBER THEOREM DIRICHLET PRIME NUMBER THEOREM JING MIAO Abstract. In number theory, the rime number theory describes the asymtotic distribution of rime numbers. We all know that there are infinitely many rimes,but how

More information

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

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

More information

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

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

More information

Recent progress in additive prime number theory

Recent progress in additive prime number theory Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns in

More information

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.

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

More information

1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft 2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

More information

Primality - Factorization

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.

More information

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

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

More information

12 Greatest Common Divisors. The Euclidean Algorithm

12 Greatest Common Divisors. The Euclidean Algorithm Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

More information

Stochastic Derivation of an Integral Equation for Probability Generating Functions

Stochastic Derivation of an Integral Equation for Probability Generating Functions Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3,. 157 163 RGN Publications htt://www.rgnublications.com Stochastic Derivation of an Integral Equation for Probability Generating

More information

2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. 2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

More information

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens.

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens. Pinhole Otics Science, at bottom, is really anti-intellectual. It always distrusts ure reason and demands the roduction of the objective fact. H. L. Mencken (1880-1956) OBJECTIVES To study the formation

More information

SMMG December 2 nd, 2006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. Fun Fibonacci Facts

SMMG December 2 nd, 2006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. Fun Fibonacci Facts SMMG December nd, 006 Dr. Edward Burger (Williams College) Discovering Beautiful Patterns in Nature and Number. The Fibonacci Numbers Fun Fibonacci Facts Examine the following sequence of natural numbers.

More information

arxiv:math/0412079v2 [math.nt] 2 Mar 2006

arxiv:math/0412079v2 [math.nt] 2 Mar 2006 arxiv:math/0412079v2 [math.nt] 2 Mar 2006 Primes Generated by Recurrence Sequences Graham Everest, Shaun Stevens, Duncan Tamsett, and Tom Ward 1st February 2008 1 MERSENNE NUMBERS AND PRIMITIVE PRIME DIVISORS.

More information

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13 COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

More information

4.2 Euclid s Classification of Pythagorean Triples

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

More information

SOLUTIONS FOR PROBLEM SET 2

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

More information