Number Theory and Its History - Brossura

Ore, Oystein

Brossura

ISBN 10: 0486656209 ISBN 13: 9780486656205

Casa editrice: Dover Pubns, 2003

"A very valuable addition to any mathematical library." — School Science and Math
This book, written by a prominent mathematician and Sterling Professor of Mathematics at Yale, differs from most other books on number theory in two important ways: first, it presents the principal ideas and methods of number theory within a historical and cultural framework, making the subject more tangible and easily grasped. Second, the material requires substantially less mathematical background than many comparable texts. Technical complications and mathematical requirements have been kept to a minimum in order to make the book as accessible as possible to readers with limited mathematical knowledge. For the majority of the book, a basic knowledge of algebra will suffice.
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.

Contenuti:

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 numerals Chapter 2. Properties of Numbers. Division 1. Number theory and numerology 2. Multiples and divisors 3. Division and remainders 4. Number systems 5. Binary number systems Chapter 3. Euclid's Algorism 1. Greatest common divisor. Euclid's algorism 2. The division lemma 3. Least 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 and 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. AI-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

Product Description:

Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography.