## Number Theory and Its History"A very valuable addition to any mathematical library." — In developing the importance and meaning of number theory in the history of mathematics, Professor Ore documents the contributions of a host of history's greatest mathematicians: Diophantos, Euclid, Fibonacci, Euler, Fermat, Mersenne, Gauss, and many more, showing how these thinkers evolved the major outlines of number theory. Topics covered include counting and recording of numbers, the properties of numbers, prime numbers, the Aliquot parts, indeterminate problems, theory of linear indeterminate problems, Diophantine problems, congruences, analysis of congruences, Wilson's Theorem, Euler's Theorem, theory of decimal expansions, the converse of Fermat's Theorem, and the classical construction problems. Based on a course the author gave for a number of years at Yale, this book covers the essentials of number theory with a clarity and avoidance of abstruse mathematics that make it an ideal resource for undergraduates or for amateur mathematicians. It has even been recommended for self-study by gifted high school students. In short, Number Theory and Its History offers an unusually interesting and accessible presentation of one of the oldest and most fascinating provinces of mathematics. This inexpensive paperback edition will be a welcome addition to the libraries of students, mathematicians, and any math enthusiast. |

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### Contents

Counting and Recording of Numbers 1 Numbers and counting | 1 |

The number systems | 2 |

Large numbers | 4 |

Finger numbers | 5 |

Recordings of numbers | 6 |

Writing of numbers | 8 |

Calculations | 14 |

Positional numeral systems | 16 |

Indeterminate Problems | 116 |

Problems with two unknowns | 124 |

Problems with several unknowns | 131 |

Chapter 7 Theory of Linear Indeterminate Problems | 142 |

Linear indeterminate equations in several unknowns | 153 |

Diophantine Problems | 165 |

Diophantos of Alexandria 17 | 179 |

AlKarkhi and Leonardo Pisano | 185 |

HinduArabic numerals | 19 |

Properties of Numbers Division 1 Number theory and numerology | 25 |

Multiples and divisors | 28 |

Division and remainders | 30 |

Number systems | 34 |

Binary number systems | 37 |

Greatest common divisor Euclids algorism | 41 |

The division lemma | 44 |

Least common multiple | 45 |

Greatest common divisor and least common multiple for several numbers | 47 |

Prime Numbers 1 Prime numbers and the prime factori2ation theorem | 50 |

Determination of prime factors | 52 |

Factor tables | 53 |

Eulers factori2ation method | 59 |

Mersenne and Fermat primes | 69 |

The distribution of primes | 75 |

The Aliquot Parts | 86 |

Amicable numbers | 96 |

Eulers function | 109 |

From Diophantos to Fermat | 194 |

Formats last theorem | 203 |

Operations with congruences 2l | 216 |

Casting out nines | 225 |

Analysis of Congruences | 234 |

Simultaneous congruences and the Chinese remainder theorem | 240 |

Further study of algebraic congruences 241 | 249 |

Wilsons Theorem and Its Consequences | 259 |

Representations of numbers as the sum of two squares | 267 |

Fermats theorem | 277 |

Primitive roots for primes | 284 |

Universal exponents | 290 |

Number theory and the splicing of telephone cables | 302 |

Decimal fractions | 311 |

Chapter 14 The Converse of Fermats Theorem | 326 |

The classical construction problems | 340 |

Supplement | 359 |

General Name Index | 361 |