Cover image for A friendly introduction to number theory
A friendly introduction to number theory
Silverman, Joseph H., 1955-
Personal Author:
Second edition.
Publication Information:
Upper Saddle River, N.J. : Prentice Hall, [2001]

Physical Description:
vii, 386 pages : illustrations ; 24 cm
Subject Term:
Format :


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QA241 .S497 2001 Adult Non-Fiction Non-Fiction Area

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This introductory text is designed to entice non-math focused individuals into learning some mathematics, while teaching them to think mathematically. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. Pythagorean Triples, Linear Equations and the Greatest Common Divisor, Factorization and the Fundamental Theorem of Arithmetic, Congruences, Mersenne Primes, Squares Modulo "p," Quadratic Reciprocity, Pell's Equation, Diophantine Approximation, Irrational Numbers and Transcendental Numbers, Sums of Powers, Binomial Coefficients and Pascal's Triangle, Elliptic Curves and Fermat's Last Theorem. For individuals with limited math experience who are interested in number theory.

Table of Contents

Prefacep. v
Introductionp. 1
Chapter 1. What Is Number Theory?p. 7
Chapter 2. Pythagorean Triplesp. 13
Chapter 3. Pythagorean Triples and the Unit Circlep. 19
Chapter 4. Sums of Higher Powers and Fermat's Last Theoremp. 23
Chapter 5. Divisibility and the Greatest Common Divisorp. 27
Chapter 6. Linear Equations and the Greatest Common Divisorp. 34
Chapter 7. Factorization and the Fundamental Theorem of Arithmeticp. 43
Chapter 8. Congruencesp. 52
Chapter 9. Congruences, Powers, and Fermat's Little Theoremp. 58
Chapter 10. Congruences, Powers, and Euler's Formulap. 64
Chapter 11. Euler's Phi Functionp. 68
Chapter 12. Prime Numbersp. 75
Chapter 13. Counting Primesp. 82
Chapter 14. Mersenne Primesp. 88
Chapter 15. Mersenne Primes and Perfect Numbersp. 92
Chapter 16. Powers Modulo m and Successive Squaringp. 102
Chapter 17. Computing k[superscript th] Roots Modulo mp. 109
Chapter 18. Powers, Roots, and "Unbreakable" Codesp. 113
Chapter 19. Euler's Phi Function and Sums of Divisorsp. 120
Chapter 20. Powers Modulo p and Primitive Rootsp. 125
Chapter 21. Primitive Roots and Indicesp. 135
Chapter 22. Squares Modulo pp. 142
Chapter 23. Is - 1 a Square Modulo p? Is 2?p. 150
Chapter 24. Quadratic Reciprocityp. 161
Chapter 25. Which Primes Are Sums of Two Squares?p. 172
Chapter 26. Which Numbers Are Sums of Two Squares?p. 184
Chapter 27. The Equation X[superscript 4] + Y[superscript 4] = Z[superscript 4]p. 190
Chapter 28. Square-Triangular Numbers Revisitedp. 193
Chapter 29. Pell's Equationp. 203
Chapter 30. Diophantine Approximationp. 208
Chapter 31. Diophantine Approximation and Pell's Equationp. 217
Chapter 32. Primality Testing and Carmichael Numbersp. 225
Chapter 33. Number Theory and Imaginary Numbersp. 236
Chapter 34. The Gaussian Integers and Unique Factorizationp. 250
Chapter 35. Irrational Numbers and Transcendental Numbersp. 267
Chapter 36. Binomial Coefficients and Pascal's Trianglep. 283
Chapter 37. Fibonacci's Rabbits and Linear Recurrence Sequencesp. 294
Chapter 38. Generating Functionsp. 306
Chapter 39. Sums of Powersp. 317
Chapter 40. Cubic Curves and Elliptic Curvesp. 328
Chapter 41. Elliptic Curves with Few Rational Pointsp. 341
Chapter 42. Points on Elliptic Curves Modulo pp. 347
Chapter 43. Torsion Collections Modulo p and Bad Primesp. 359
Chapter 44. Defect Bounds and Modularity Patternsp. 363
Chapter 45. Elliptic Curves and Fermat's Last Theoremp. 369
Further Readingp. 371
Appendix A. Factorization of Small Composite Integersp. 372
Appendix B. A List of Primesp. 374
Indexp. 377