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Friendly Introduction to Number Theory, A (Classic Version), 4th edition

Published by Pearson (February 13, 2017) © 2018

  • Joseph H. Silverman

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For 1-semester undergraduate courses in Elementary Number Theory.

A modern classic

A Friendly Introduction to Number Theory, 4th Edition introduces 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 analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

Hallmark features of this title

  • 50 short chapters provide flexibility and options for instructors and students. A flowchart of chapter dependencies is included.
  • 5 basic steps are emphasized throughout the text to help readers develop a robust thought process: Experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.
  • RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured, showing the real-life applications of mathematics.

New and updated features of this title

  • Many new exercises appear throughout the text.
  • Content updates throughout include:
    • A new chapter on mathematical induction (Chapter 26).
    • Some material on proof by contradiction has been moved forward to Chapter 8.
    • 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).
    • 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.

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 kth 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 X4 + Y4 = Z4
  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
  47. The Topsy-Turvey World of Continued Fractions [online]
  48. Continued Fractions, Square Roots, and Pell's Equation [online]
  49. Generating Functions [online]
  50. Sums of Powers [online]

Further Reading

Index

A. Factorization of Small Composite Integers [online]

B. A List of Primes [online]

About our author

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 7 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.

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