Number Theory: A Historical Approach

Princeton University Press, 26 de des. 2013 - 592 pàgines

The natural numbers have been studied for thousands of years, yet most undergraduate textbooks present number theory as a long list of theorems with little mention of how these results were discovered or why they are important. This book emphasizes the historical development of number theory, describing methods, theorems, and proofs in the contexts in which they originated, and providing an accessible introduction to one of the most fascinating subjects in mathematics.

Written in an informal style by an award-winning teacher, Number Theory covers prime numbers, Fibonacci numbers, and a host of other essential topics in number theory, while also telling the stories of the great mathematicians behind these developments, including Euclid, Carl Friedrich Gauss, and Sophie Germain. This one-of-a-kind introductory textbook features an extensive set of problems that enable students to actively reinforce and extend their understanding of the material, as well as fully worked solutions for many of these problems. It also includes helpful hints for when students are unsure of how to get started on a given problem.

• Uses a unique historical approach to teaching number theory

• Features numerous problems, helpful hints, and fully worked solutions

• Discusses fun topics like Pythagorean tuning in music, Sudoku puzzles, and arithmetic progressions of primes

• Includes an introduction to Sage, an easy-to-learn yet powerful open-source mathematics software package

• Ideal for undergraduate mathematics majors as well as non-math majors

• Digital solutions manual (available only to professors)

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Continguts

 Number Theory Begins 1 Square Numbers 7 Arithmetic Progressions 14 Euclid 26 The Axiomatic Method 33 Unique Factorization 41 Problems 47 Divisibility 59
 Goldbachs Conjecture 296 Sophie Germain 307 Problems 317 Fibonacci Numbers 324 Fibonacci Numbers in Nature 331 Cryptography 364 Divisibility Tests 379 Partitions 433

 The Fundamental Theorem of Arithmetic 68 Diophantus 90 Fermat 116 Congruences 165 Euler and Lagrange 188 Gauss 227 Primes I 258 Is n Prime? 267 Mersenne Primes 273 Primes II 285
 Problems 463 Solutions to Selected Problems 481 1 488 12 507 9 527 3 545 Brief Introduction to Sage 559 Pronunciation Guide 569 309 574 Copyright

Sobre l'autor (2013)

John J. Watkins is professor emeritus of mathematics at Colorado College. His books include Across the Board: The Mathematics of Chessboard Problems (Princeton), Topics in Commutative Ring Theory (Princeton), Graphs: An Introductory Approach, and Combinatorics: Ancient and Modern.