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### Summary

### Summary

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

Preface | p. v |

Introduction | p. 1 |

Chapter 1. What Is Number Theory? | p. 7 |

Chapter 2. Pythagorean Triples | p. 13 |

Chapter 3. Pythagorean Triples and the Unit Circle | p. 19 |

Chapter 4. Sums of Higher Powers and Fermat's Last Theorem | p. 23 |

Chapter 5. Divisibility and the Greatest Common Divisor | p. 27 |

Chapter 6. Linear Equations and the Greatest Common Divisor | p. 34 |

Chapter 7. Factorization and the Fundamental Theorem of Arithmetic | p. 43 |

Chapter 8. Congruences | p. 52 |

Chapter 9. Congruences, Powers, and Fermat's Little Theorem | p. 58 |

Chapter 10. Congruences, Powers, and Euler's Formula | p. 64 |

Chapter 11. Euler's Phi Function | p. 68 |

Chapter 12. Prime Numbers | p. 75 |

Chapter 13. Counting Primes | p. 82 |

Chapter 14. Mersenne Primes | p. 88 |

Chapter 15. Mersenne Primes and Perfect Numbers | p. 92 |

Chapter 16. Powers Modulo m and Successive Squaring | p. 102 |

Chapter 17. Computing k[superscript th] Roots Modulo m | p. 109 |

Chapter 18. Powers, Roots, and "Unbreakable" Codes | p. 113 |

Chapter 19. Euler's Phi Function and Sums of Divisors | p. 120 |

Chapter 20. Powers Modulo p and Primitive Roots | p. 125 |

Chapter 21. Primitive Roots and Indices | p. 135 |

Chapter 22. Squares Modulo p | p. 142 |

Chapter 23. Is - 1 a Square Modulo p? Is 2? | p. 150 |

Chapter 24. Quadratic Reciprocity | p. 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 Revisited | p. 193 |

Chapter 29. Pell's Equation | p. 203 |

Chapter 30. Diophantine Approximation | p. 208 |

Chapter 31. Diophantine Approximation and Pell's Equation | p. 217 |

Chapter 32. Primality Testing and Carmichael Numbers | p. 225 |

Chapter 33. Number Theory and Imaginary Numbers | p. 236 |

Chapter 34. The Gaussian Integers and Unique Factorization | p. 250 |

Chapter 35. Irrational Numbers and Transcendental Numbers | p. 267 |

Chapter 36. Binomial Coefficients and Pascal's Triangle | p. 283 |

Chapter 37. Fibonacci's Rabbits and Linear Recurrence Sequences | p. 294 |

Chapter 38. Generating Functions | p. 306 |

Chapter 39. Sums of Powers | p. 317 |

Chapter 40. Cubic Curves and Elliptic Curves | p. 328 |

Chapter 41. Elliptic Curves with Few Rational Points | p. 341 |

Chapter 42. Points on Elliptic Curves Modulo p | p. 347 |

Chapter 43. Torsion Collections Modulo p and Bad Primes | p. 359 |

Chapter 44. Defect Bounds and Modularity Patterns | p. 363 |

Chapter 45. Elliptic Curves and Fermat's Last Theorem | p. 369 |

Further Reading | p. 371 |

Appendix A. Factorization of Small Composite Integers | p. 372 |

Appendix B. A List of Primes | p. 374 |

Index | p. 377 |