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

  • Joseph H. Silverman

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A Friendly Introduction to Number Theory, 4th Edition introduces you 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, you are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for undergraduates 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.

Published by Pearson (May 24th 2023) - Copyright © 2023

ISBN-13: 9780137981175

Subject: Advanced Math

Category: Number Theory

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]