# Book: An Introduction to the Theory of Numbers (Moser) - Mathematics

This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.

• Chapter 1. Compositions and Partitions
• Chapter 2. Arithmetic Functions
• Chapter 3. Distribution of Primes
• Chapter 4. Irrational Numbers
• Chapter 5. Congruences
• Chapter 6. Diophantine Equations
• Chapter 7. Combinatorial Number Theory
• Chapter 8. Geometry of Numbers

This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions Arithmetic Functions Distribution of Primes Irrational Numbers Congruences Diophantine Equations Combinatorial Number Theory and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.

## An Introduction to the Theory of Numbers

An Introduction to the Theory of Numbers is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright.

The book grew out of a series of lectures by Hardy and Wright and was first published in 1938.

The third edition added an elementary proof of the prime number theorem, and the sixth edition added a chapter on elliptic curves.

• Bell, E. T. (1939), "Book Review: An Introduction to the Theory of Numbers", Bulletin of the American Mathematical Society, 45 (7): 507–509, doi: 10.1090/S0002-9904-1939-07025-0 , ISSN0002-9904
• Hardy, Godfrey Harold Wright, E. M. (1938), An introduction to the theory of numbers. (First ed.), Oxford: Clarendon Press, JFM64.0093.03, Zbl0020.29201
• Hardy, Godfrey Harold Wright, E. M. (1954) , An introduction to the theory of numbers (Third ed.), Oxford, at the Clarendon Press, MR0067125
• Hardy, Godfrey Harold Wright, E. M. (1979) , An introduction to the theory of numbers (Fifth ed.), The Clarendon Press Oxford University Press, ISBN978-0-19-853171-5 , MR0568909
• Hardy, Godfrey Harold Wright, E. M. (2008) , Heath-Brown, D. R. Silverman, J. H. (eds.), An introduction to the theory of numbers (Sixth ed.), Oxford University Press, ISBN978-0-19-921986-5 , MR2445243

## An Introduction to the Theory of Numbers "If I could bring only one book with me to a desert island, it would be [some other book] if I thought I would be rescued. It would be G.H. Hardy&aposs Theory of Numbers if I knew I was never coming back."

Sold. I read a two sentence review of this once that has really stuck with me. It went along the lines of

"If I could bring only one book with me to a desert island, it would be [some other book] if I thought I would be rescued. It would be G.H. Hardy's Theory of Numbers if I knew I was never coming back."

Introduction to the Theory of Numbers by Godfrey Harold Hardy is more sturdy than the other book by him that I had read recently. It is also significantly longer. While E. M. Wright also went and wrote some things for this book, he wasn’t included on the spine of the book, so I forgot about him. The opening section of the book states that it arose out of series of lectures given at Oxford, Cambridge, and other Universities. Given that, it is not a systematic treatment of the subject, though it d Introduction to the Theory of Numbers by Godfrey Harold Hardy is more sturdy than the other book by him that I had read recently. It is also significantly longer. While E. M. Wright also went and wrote some things for this book, he wasn’t included on the spine of the book, so I forgot about him. The opening section of the book states that it arose out of series of lectures given at Oxford, Cambridge, and other Universities. Given that, it is not a systematic treatment of the subject, though it does attempt to touch upon all of the facets of Number Theory.

This book starts out by discussing the terminology and symbols used. It is divided into twenty-four chapters starting out with Prime Numbers. The chapters go like this:

(1. The Series of Primes (1)
(2. The Series of Primes (2)
(3. Farey Series and a Theorem of Minkowski
(4. Irrational Numbers
(5. Congruences and Residues
(6. Fermat’s Theorem and its Consequences
(7. General Properties of Congruences
(8. Congruences to Composite Moduli
(9. The Representation of Numbers by Decimals
(10. Continued Fractions
(11. Approximation of Irrationals by Rationals
(12. The Fundamental Theorem of Arithmetic in k(1), k(i), and k(ρ)
(13. Some Diophantine Equations
(16. The Arithmetical Functions ϕ(n), μ(n), d(n), σ(n), r(n)
(17. Generating Functions of Arithmetical Functions
(18. The Order of Magnitude of Arithmetical Functions
(19. Partitions
(20. The Representation of a Number by Two or Four Squares
(21. Representation by Cubes and Higher Powers
(22. The Series of Primes (3)
(23. Kronecker’s Theorem
(24. Some More Theorems of Minkowski

Most of the chapters are self-explanatory. Some of them are rather opaque at first glance. Take chapter 19 for example, it is called Partitions. What exactly is a partition? Looking into it tells you that a partition is a way to show a number using any number of positive integral parts.

I especially liked the chapters on modular arithmetic since that is something I never really learned in school for some reason. This particular version was written in 1938 and I don't know what edition it is. . more

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Godfrey H. Hardy, Edward M. Wright

New - Hardcover
Condition: New

Hardback. Condition: New. 6th Revised edition. Language: English. Brand new Book. An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D.R. Heath-Brown this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones anddevelopments in number theory. Updates include a chapter by J.H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem - a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid readerThe text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists. ## A Friendly Introduction to Number Theory

A Friendly Introduction to Number Theory is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while at the same time teaching them how to think mathematically. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures. Only then are theorems proved, with the emphasis on methods of proof rather than on specific results. Starting with nothing more than basic high school algebra, the reader is gradually led to the point of producing their own conjectures and proofs, as well as getting some glimpses at the frontiers of current mathematical research.

Instructors : To receive an evaluation copy of A Friendly Introduction to Number Theory , send an email request to:
Stacey Sveum, Marketing Manager, Prentice-Hall.

Click on the links for the following material.

• Chapters 1&ndash6
• Chapter 1: What Is Number Theory?
• Chapter 2: Pythagorean Triples
• Chapter 3: Pythagorean Triples and the Unit Circle
• Chapter 4: Sums of Higher Powers and Fermat's Last Theorem
• Chapter 5: Divisibility and the Greatest Common Divisor
• Chapter 6: Linear Equations and the Greatest Common Divisor
• Chapter 47: The Topsy-Turvy World of Continued Fractions
• Chapter 48: Continued Fractions and Pell's Equation
• Chapter 49: Generating Functions
• Chapter 50: Sums of Powers
• Appendix A: Factorization of Small Composite Integers
• Appendix B: A List of primes
• Exercise 18.4
• Exercise 18.7
• Exercise 19.8
• Exercise 22.7

Errata for the 4th edition. (Errata for the 3rd edition is also available.)

• There is a new chapter on mathematical induction (Chapter 26).
• Some material on proof by contradiction has been moved forward to Chapter 8. It is used in the proof that a polynomial of degree d has at most d roots modulo p . This fact is then used in place of primitive roots as a tool to prove Euler's quadratic residue formula in Chapter 21. (In earlier editions, primitive roots were used for this proof.)
• The chapters on primitive roots (Chapters 28&ndash29) have been moved to follow the chapters on quadratic reciprocity and sums of squares (Chapters 20&ndash25). The rationale for this change is the author's experience that students find the Primitive Root Theorem to be among the most difficult in the book. The new order allows the instructor to cover quadratic reciprocity first, and to omit primitive roots entirely if desired.
• Chapter 22 now includes a proof of part of quadratic reciprocity for Jacobi symbols, with the remaining parts included as exercises.
• Quadratic reciprocity is now proved in full. The proofs for (-1| p ) and (2| p ) remain as before in Chapter 21, and there is a new chapter (Chapter 23) that gives Eisenstein's proof for ( p | q )( q | p ). Chapter 23 is significantly more difficult than the chapters that precede it, and it may be omitted without affecting the subsequent chapters.
• As an application of primitive roots, Chapter 28 discusses the construction of Costas arrays.
• Chapter 39 includes a proof that the period of the Fibonacci sequence modulo p divides p &ndash1 when p is congruent to 1 or 4 modulo 5.
• There are many new exercises scattered throughout the text.
• A flowchart giving chapter dependencies is included on page ix.
• Number theory is a vast and sprawling subject, and over the years this book has acquired many new chapters. In order to keep the length of this edition to a reasonable size, Chapters 47&ndash50 have been removed from the printed version of the book. These omitted chapters are freely available by clicking the following link: Chapters 47&ndash50. The online chapters are included in the index.

Exercise 18.4 Here are two longer messages to decode if you like to use computers.
(a) You have been sent the following message:

5272281348, 21089283929, 3117723025, 26844144908, 22890519533,
26945939925, 27395704341, 2253724391, 1481682985, 2163791130,
13583590307, 5838404872, 12165330281, 28372578777, 7536755222.

It has been encoded using

p = 187963, q = 163841, m = pq = 30796045883, and k = 48611.

(b) You intercept the following message, which you know has been encoded using the modulus

m = 956331992007843552652604425031376690367 and exponent k = 12398737.

Break the code and decipher the message.

821566670681253393182493050080875560504,
87074173129046399720949786958511391052,
552100909946781566365272088688468880029,
491078995197839451033115784866534122828,
172219665767314444215921020847762293421.

Exercise 18.7 Write a computer program implementing one of the factorization methods that you studied in the previous exercise, such as Pollard's &rho method, Pollard's p -1 method, or the quadratic sieve. Use your program to factor the following numbers.
(a) 47386483629775753
(b) 1834729514979351371768185745442640443774091

Exercise 19.8 Program the Rabin-Miller test with multiprecision integers and use it to investigate which of the following numbers are composite.
(a) 155196355420821961
(b) 155196355420821889
(c) 285707540662569884530199015485750433489
(d) 285707540662569884530199015485751094149

Exercise 22.7 For this exercise, use the ElGamal cryptosystem described in Exercise 22.6.
(a) Bob wants to use Alice's public key a = 22695 for the prime p = 163841 and base g = 3 to send her the message m = 39828. He chooses to use the random number r = 129381. Compute the encrypted message ( e 1 , e 2 ) he should send to Alice.
(b) Suppose that Bob sends the same message to Alice, but he chooses a different value for r . Will the encrypted message be the same?
(c) Alice has chosen the secret key k = 278374 for the prime p = 380803 and the base g = 2. She receives a message (consisting of three message blocks)

(61745, 206881), (255836, 314674), (108147, 350768)

from Bob. Decrypt the message and convert it to letters using the number-to-letter conversion table in Chapter 18.

Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica . Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively.

This book describes many applications including modern applications in cryptography it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.

## An Introduction To The Geometry Of Numbers

Author by : J.W.S. Cassels
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Publisher by : Springer Science & Business Media
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Description : From the reviews: "A well-written, very thorough account . Among the topics are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs some applications to number theory excellent bibliographical references." The American Mathematical Monthly

## Book: An Introduction to the Theory of Numbers (Moser) - Mathematics

After distributing hundreds of thousands of copies of Elias Zakon's Basic Concepts of Mathematics and the award-winning Mathematical Analysis I, we are delighted to announce the release of the final text in the Zakon Series on Mathematical Analysis, Mathematical Analysis II, which presents the material of a typical graduate course on real analysis. Read more about this book.

Also of Note: Those readers with an interest in number theory should see Leo Moser's An Introduction to the Theory of Numbers.

Online Math: We have added a section with links to online mathematics materials. Most of the links in this section were originally collected by Alex Stefanov.

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