Softcover ISBN:  9781470470272 
Product Code:  AMSTEXT/61 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470472580 
Product Code:  AMSTEXT/61.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Softcover ISBN:  9781470470272 
eBook: ISBN:  9781470472580 
Product Code:  AMSTEXT/61.B 
List Price:  $178.00 $133.50 
MAA Member Price:  $160.20 $120.15 
AMS Member Price:  $142.40 $106.80 
Softcover ISBN:  9781470470272 
Product Code:  AMSTEXT/61 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470472580 
Product Code:  AMSTEXT/61.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Softcover ISBN:  9781470470272 
eBook ISBN:  9781470472580 
Product Code:  AMSTEXT/61.B 
List Price:  $178.00 $133.50 
MAA Member Price:  $160.20 $120.15 
AMS Member Price:  $142.40 $106.80 

Book DetailsPure and Applied Undergraduate TextsVolume: 61; 2023; 442 ppMSC: Primary 00; 03; 05; 11; 97; Secondary 68
Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and outofdate pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementarysounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles.
ReadershipAppropriate for a course serving as a transition to advanced mathematics and for undergraduate students interested in mathematical reasoning and an introduction to proofs.

Table of Contents

Cover

Title page

Contents

Preface

Philosophy about learning and teaching

Content of this book

Style of this book

Problem solving

LaTeX

Origins

Further reading

Acknowledgments

Notations and Symbols

Chapter 1. Evens, Odds, and Primes: A Taste of Number Theory

1.1. A first excursion into prime numbers

1.2. Even and odd integers

1.3. Calculating primes and the sieve of Eratosthenes

1.4. Division

1.5. Greatest common divisor

1.6. Statement of prime factorization

1.7*. Perfect numbers

1.8*. One of the Mersenne conjectures

1.9*. Twin primes: An excursion into the unknown

1.10*. Goldbach’s conjecture

1.11. Hints and partial solutions for the exercises

Chapter 2. Mathematical Induction

2.1. Mathematical induction

2.2. Rates of growth of functions

2.3. Sums of powers of the first 𝑛 positive integers

2.4. Strong mathematical induction

2.5. Fibonacci numbers

2.6. Recursive definitions

2.7. Arithmetic and algebraic equalities and inequalities

2.8. Hints and partial solutions for the exercises

Chapter 3. Logic: Implications, Contrapositives, Contradictions, and Quantifiers

3.1. The need for rigor

3.2. Statements

3.3. Truth teller and liar riddle: Asking the right question

3.4*. Logic puzzles

3.5. Logical connectives

3.6. Implications

3.7. Contrapositive

3.8. Proof by contradiction

3.9. Pythagorean triples

3.10. Quantifiers

3.11. Hints and partial solutions for the exercises

Chapter 4. The Euclidean Algorithm and Its Consequences

4.1. The Division Theorem

4.2. There are an infinite number of primes

4.3. The Euclidean algorithm

4.4. Consequences of the Division Theorem

4.5. Solving linear Diophantine equations

4.6. “Practical” applications of solving linear Diophantine equations (wink \smiley)

4.7*. (Polynomial) Diophantine equations

4.8. The Fundamental Theorem of Arithmetic

4.9. The least common multiple

4.10. Residues modulo an odd prime

4.11. Appendix

4.12. Hints and partial solutions for the exercises

Chapter 5. Sets and Functions

5.1. Basics of set theory

5.2. Cartesian products of sets

5.3. Functions and their properties

5.4. Types of functions: Injections, surjections, and bijections

5.5. Arbitrary unions, intersections, and cartesian products

5.6*. Universal properties of surjections and injections

5.7. Hints and partial solutions for the exercises

Chapter 6. Modular Arithmetic

6.1. Multiples of 3 and 9 and the digits of a number in base 10

6.2. Congruence modulo 𝑚

6.3. Inverses, coprimeness, and congruence

6.4. Congruence and multiplicative cancellation

6.5*. Fun congruence facts

6.6. Solving linear congruence equations

6.7*. The Chinese Remainder Theorem

6.8. Quadratic residues

6.9. Fermat’s Little Theorem

6.10*. Euler’s totient function and Euler’s Theorem

6.11*. An application of Fermat’s Little Theorem: The RSA algorithm

6.12*. The Euclid–Euler Theorem characterizing even perfect numbers

6.13*. Twin prime pairs

6.14. Chameleons roaming around in a zoo

6.15. Hints and partial solutions for the exercises

Chapter 7. Counting Finite Sets

7.1. The addition principle

7.2. Cartesian products and the multiplication principle

7.3. The inclusionexclusion principle

7.4. Binomial coefficients and the Binomial Theorem

7.5. Counting functions

7.6. Counting problems

7.7*. Using the idea of a bijection

7.8. Hints and partial solutions for the exercises

Chapter 8. Congruence Class Arithmetic, Groups, and Fields

8.1. Congruence classes modulo 𝑚

8.2. Inverses of congruence classes

8.3. Reprise of the proof of Fermat’s Little Theorem

8.4. Equivalence relations, equivalence classes, and partitions

8.5. Elementary abstract algebra

8.6. Rings, principal ideal domains, and all that

8.7. Fields

8.8*. Quadratic residues and the law of quadratic reciprocity

8.9. Hints and partial solutions for the exercises

Bibliography

Index

Back Cover


Additional Material

Reviews

"Introduction to Proof Through Number Theory" is a fun book that introduces students to mathematical thinking and proof writing through a deeper understanding of elementary concepts in Number Theory. The book's informal style with humor, games, puzzles, and connections to popular culture with references to music, art, movies and literature lightens up the material without compromising rigor.
Hema Gopalakrishnan, MAA Reviews


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and outofdate pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementarysounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles.
Appropriate for a course serving as a transition to advanced mathematics and for undergraduate students interested in mathematical reasoning and an introduction to proofs.

Cover

Title page

Contents

Preface

Philosophy about learning and teaching

Content of this book

Style of this book

Problem solving

LaTeX

Origins

Further reading

Acknowledgments

Notations and Symbols

Chapter 1. Evens, Odds, and Primes: A Taste of Number Theory

1.1. A first excursion into prime numbers

1.2. Even and odd integers

1.3. Calculating primes and the sieve of Eratosthenes

1.4. Division

1.5. Greatest common divisor

1.6. Statement of prime factorization

1.7*. Perfect numbers

1.8*. One of the Mersenne conjectures

1.9*. Twin primes: An excursion into the unknown

1.10*. Goldbach’s conjecture

1.11. Hints and partial solutions for the exercises

Chapter 2. Mathematical Induction

2.1. Mathematical induction

2.2. Rates of growth of functions

2.3. Sums of powers of the first 𝑛 positive integers

2.4. Strong mathematical induction

2.5. Fibonacci numbers

2.6. Recursive definitions

2.7. Arithmetic and algebraic equalities and inequalities

2.8. Hints and partial solutions for the exercises

Chapter 3. Logic: Implications, Contrapositives, Contradictions, and Quantifiers

3.1. The need for rigor

3.2. Statements

3.3. Truth teller and liar riddle: Asking the right question

3.4*. Logic puzzles

3.5. Logical connectives

3.6. Implications

3.7. Contrapositive

3.8. Proof by contradiction

3.9. Pythagorean triples

3.10. Quantifiers

3.11. Hints and partial solutions for the exercises

Chapter 4. The Euclidean Algorithm and Its Consequences

4.1. The Division Theorem

4.2. There are an infinite number of primes

4.3. The Euclidean algorithm

4.4. Consequences of the Division Theorem

4.5. Solving linear Diophantine equations

4.6. “Practical” applications of solving linear Diophantine equations (wink \smiley)

4.7*. (Polynomial) Diophantine equations

4.8. The Fundamental Theorem of Arithmetic

4.9. The least common multiple

4.10. Residues modulo an odd prime

4.11. Appendix

4.12. Hints and partial solutions for the exercises

Chapter 5. Sets and Functions

5.1. Basics of set theory

5.2. Cartesian products of sets

5.3. Functions and their properties

5.4. Types of functions: Injections, surjections, and bijections

5.5. Arbitrary unions, intersections, and cartesian products

5.6*. Universal properties of surjections and injections

5.7. Hints and partial solutions for the exercises

Chapter 6. Modular Arithmetic

6.1. Multiples of 3 and 9 and the digits of a number in base 10

6.2. Congruence modulo 𝑚

6.3. Inverses, coprimeness, and congruence

6.4. Congruence and multiplicative cancellation

6.5*. Fun congruence facts

6.6. Solving linear congruence equations

6.7*. The Chinese Remainder Theorem

6.8. Quadratic residues

6.9. Fermat’s Little Theorem

6.10*. Euler’s totient function and Euler’s Theorem

6.11*. An application of Fermat’s Little Theorem: The RSA algorithm

6.12*. The Euclid–Euler Theorem characterizing even perfect numbers

6.13*. Twin prime pairs

6.14. Chameleons roaming around in a zoo

6.15. Hints and partial solutions for the exercises

Chapter 7. Counting Finite Sets

7.1. The addition principle

7.2. Cartesian products and the multiplication principle

7.3. The inclusionexclusion principle

7.4. Binomial coefficients and the Binomial Theorem

7.5. Counting functions

7.6. Counting problems

7.7*. Using the idea of a bijection

7.8. Hints and partial solutions for the exercises

Chapter 8. Congruence Class Arithmetic, Groups, and Fields

8.1. Congruence classes modulo 𝑚

8.2. Inverses of congruence classes

8.3. Reprise of the proof of Fermat’s Little Theorem

8.4. Equivalence relations, equivalence classes, and partitions

8.5. Elementary abstract algebra

8.6. Rings, principal ideal domains, and all that

8.7. Fields

8.8*. Quadratic residues and the law of quadratic reciprocity

8.9. Hints and partial solutions for the exercises

Bibliography

Index

Back Cover

"Introduction to Proof Through Number Theory" is a fun book that introduces students to mathematical thinking and proof writing through a deeper understanding of elementary concepts in Number Theory. The book's informal style with humor, games, puzzles, and connections to popular culture with references to music, art, movies and literature lightens up the material without compromising rigor.
Hema Gopalakrishnan, MAA Reviews