# and Its History

## Oystein Ore

### CONTENTS

Preface

Chapter 1. Counting and Recording of Numbers

1. Numbers and counting
2. Basic number groups
3. The number systems
4. Large numbers
5. Finger numbers
6. Recordings of numbers
7. Writing of numbers
8. Calculations
9. Positional numeral systems
10. Hindu-Arabic numerak

Chapter 2. Properites of Numbers. Division.

1. Number theory and numerology
2. Multiples and divisors
3. Division and remainders
4. Number systems
5. Bimu number systems

Chapter 3. Euclid's Algorism

1. Greatest conunon divisor. Euclid's algorism
2. The division lemma
3. Umt common multiple
4. Greatest common divisor and least common multiple for several numbers

Chapter 4. Prime Numbers

1. Prime numbers and the prime factorization theorem
2. Determination of prime factors
3. Factor tables
4. Fermat's factorization method
5. Euler's factorization method
6. The sieve of Eratosthenes
7. Mersenne and Fermat primes
8. The distribution of primes

Chapter 5. The Aliquot Parts

1. The divisors of a number
2. Perfect numbers
3. Amicable numbers
4. Greatest common divisor anl least common multiple
5. Euler's function

Chapter 6. Indeterminate Problems

1. Problems and puzzles
2. Indeterminate problems
3. Problems with two unknowns
4. Problems with several unknowns

Chapter 7. Theory of Linear Indeterminate Problems

1. Theory of linear indeterminate equations with two unknowns
2. Linear indeterminate equations in several unknowns
3. Classification of systems of numbers

Chapter 8. Diophantine Problems

1. The Pythagorean triangle
2. The Plimpton Library tablet
3. Diophantos of Alexandria
4. Al-Karkhi and Leonardo Pisano
5. From Diophantos to Fermat
6. The method of infinite descent
7. Fermat's last theorem

Chapter 9. Congruences

1. The Disquisitiones arithmeticae
2. The properties of congruences
3. Residue systems
4. Operations with congruences
5. Casting out nines

Chapter 10. Analysis of Congruences

1. Algebraic congruences
2. Linear congruences
3. Simultaneous congruences and the Chinese remainder theorem
4. Further study of algebraic congruences

Chapter 11. Wilson's Theorem and Its Consequences

1. Wilson's theorem
2. Gauss's generalization of Wilson's theorem
3. Representations of numbers as the sum of two squares

Chapter 12. Euler's Theorem and Its Consequences

1. Euler's theorem
2. Fermat's theorem
3. Exponents of numbers
4. Primitive roots for primes
5. Primitive roots for powers of primes
6. Universal exponents
7. Indices
8. Number theory and the splicing of telephone cables

Chapter 13. Theory of Decimal Expansions

1. Decimal fractions
2. The properties of decimal fractions

Chapter 14. The Converse of Fermat's Theorem

1. The converse of Fermat's theorem
2. Numbers with the Fermat property

Chapter 15. The Classical Construction Problems

1. The classical construction problems
2. The construction of regular polygons
3. Examples of constructible polygons

Supplement
Bibliography
General Name Index
Subject Index 