# A Friendly Introduction to Number Theory (Classic Version) (4e)

##### Description

For one-semester undergraduate courses in Elementary Number Theory

**is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet–number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analysed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.**

*A Friendly Introduction to Number Theory*, 4th Edition##### Table of contents

Preface

Flowchart of Chapter Dependencies

Introduction

1. What Is Number Theory?

2. Pythagorean Triples

3. Pythagorean Triples and the Unit Circle

4. Sums of Higher Powers and Fermat’s Last Theorem

5. Divisibility and the Greatest Common Divisor

6. Linear Equations and the Greatest Common Divisor

7. Factorization and the Fundamental Theorem of Arithmetic

8. Congruences

9. Congruences, Powers, and Fermat’s Little Theorem

10. Congruences, Powers, and Euler’s Formula

11. Euler’s Phi Function and the Chinese Remainder Theorem

12. Prime Numbers

13. Counting Primes

14. Mersenne Primes

15. Mersenne Primes and Perfect Numbers

16. Powers Modulo *m* and Successive Squaring

17. Computing *k ^{th}* Roots Modulo

*m*

18. Powers, Roots, and “Unbreakable” Codes

19. Primality Testing and Carmichael Numbers

20. Squares Modulo p

21. Is -1 a Square Modulo *p*? Is 2?

22. Quadratic Reciprocity

23. Proof of Quadratic Reciprocity

24. Which Primes Are Sums of Two Squares?

25. Which Numbers Are Sums of Two Squares?

26. As Easy as One, Two, Three

27. Euler’s Phi Function and Sums of Divisors

28. Powers Modulo p and Primitive Roots

29. Primitive Roots and Indices

30. The Equation *X*^{4} + *Y*^{4} = *Z*^{4}

31. Square–Triangular Numbers Revisited

32. Pell’s Equation

33. Diophantine Approximation

34. Diophantine Approximation and Pell’s Equation

35. Number Theory and Imaginary Numbers

36. The Gaussian Integers and Unique Factorization

37. Irrational Numbers and Transcendental Numbers

38. Binomial Coefficients and Pascal’s Triangle

39. Fibonacci’s Rabbits and Linear Recurrence Sequences

40. Oh, What a Beautiful Function

41. Cubic Curves and Elliptic Curves

42. Elliptic Curves with Few Rational Points

43. Points on Elliptic Curves Modulo *p*

44. Torsion Collections Modulo *p* and Bad Primes

45. Defect Bounds and Modularity Patterns

46. Elliptic Curves and Fermat’s Last Theorem

##### New to this edition

There are a number of major changes in the Fourth Edition.

**Many new exercises**appear throughout the text.- A
**flowchart**giving chapter dependencies is included to help instructors choose the most appropriate mix of topics for their students. - Content Updates
- 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 28–29) have been moved to follow the chapters on quadratic reciprocity and sums of squares (Chapters 20–25). 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 −*1 when*p*is congruent to 1 or 4 modulo 5. **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, 48, 49, and 50 have been removed from the printed version of the book. These omitted chapters are freely available online at http://math.brown.edu/~jhs/frint.html or www.pearsonhighered.com/silverman. The online chapters are included in the index.

##### Features & benefits

**50 short chapters**provide flexibility and options for instructors and students. A flowchart of chapter dependencies is included in this edition.**Five basic steps**are emphasised throughout the text to help readers develop a robust thought process:- Experimentation
- Pattern recognition
- Hypothesis formation
- Hypothesis testing
- Formal proof

**RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured**, showing the real-life applications of mathematics.

##### Author biography

**Joseph H. Silverman**is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and seven books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.