17 Lectures on Fermat Numbers: From Number Theory to Geometry

ISBN-13: 9780387953328 / Angielski / Twarda / 2002 / 257 str.

M. Krizek; F. Luca; L. Somer

Opis

The pioneering work of French mathematician Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth, providing readers with an overview of the many properties of Fermat numbers and demonstrating their applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers.

Recenzje

From the reviews:

CANADIAN MATHEMATICAL SOCIETY NOTES

"It is certainly satisfying for the reader to have almost everything that is known on a particular subject collected in one place. Also, it contributes to a sense of mathematics as a unified subject to see several different areas enter, often in unexpected ways...I can...recommend it to number-theorists and other mathematicians alike; everybody will find useful and interesting information in it. In the classroom, this book might be suitable for undergraduate projects or supplementary readings in a number theory, algebra, or history of mathematics course. It will also be accessible to bright high school students and interested amateurs. In fact, Fermat's name in the title may attract such readers, and they will find a well written, interesting, and mathematically sound book..."

 

M. Krizek; F. L. Luca; and L. Somer

17 Lectures on Fermat Numbers

From Number Theory to Geometry

"The authors have brought together a wealth of material involving the Fermat numbers . . . amateurs and high-school students should also be able to profitably read this well-written book."-MATHEMATICAL REVIEWS

"This admirable book contains what must be everything that is worth knowing about Fermat numbers. ... It contains a wealth of fascinating results ... . The Foreword, by Alena Solcová, is an excellent account of Fermat's life and works. ... The authors, and the series editors, Jonathan and Peter Borwein, deserve credit and praise for producing what will be the definitive work on the subject for many years to come." (Dudley Underwood, Math DL, May, 2005)

"This book was written in celebration of the 400^th anniversary of the birth of Pierre de Fermat ... . The clear writing style and the evident enthusiasm of the authors make this book a pleasure to read or to dip into. It is not a text book, but might well inspire students at any levels, and it succeeds in making what is potentially a dry and abstract topic fresh and full of connections. It would be a valuable addition to any library." (Alison Wolff, The Australian Mathematical Society Gazette, Vol. 31 (3), 2004)

"The book is very interesting and well written - an enjoyable and very thorough collection of all the knowledge on Fermat numbers that have been accumulated during the centuries. ... Those who are interested in the history of mathematics will find answers for many questions concerning the history of Fermat numbers. This is an excellent book that can be recommended to all those interested in Fermat numbers or generally in number theory." (László Megyesi, Acta Scientiarium Mathematicarum, Vol. 69, 2003)

"In this book the authors have brought together a wealth of material involving the Fermat numbers ... . amateurs and high-school students should also be able to profitably read this well-written book. ... It has 71 illustrations in total, an extensive list of references and a good subject and name index." (Pieter Moree, Mathematical Reviews, 2002 i)

"It is certainly satisfying for the reader to have almost everything that is known on a particular subject collected in one place. ... The bibliography, with more than 350 entries, is very complete and contributes to the value of the book as reference, as do the extensive name and subject indexes. ... I can ... recommend it to number-theorists and other mathematicians alike; everybody will find useful and interesting information in it. ... a well written, interesting, and mathematically sound book ... ." (CMS Notes, Vol. 34 (8), 2002)

"Pierre de Fermat was born just over four hundred years ago, and this book is in part a celebration of his quatercentenary. ... A pleasing feature of the exposition is that as many results as possible are illustrated pictorially. ... The text is copiously illustrated with both photographs and excellent diagrams, and I can recommend the book as a whole to anyone who wants to venture into this treasure-trove of somewhat arcane number theory." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)

"The purpose of this book is to provide readers with an overview of the many properties of Fermat numbers and to demonstrate their numerous appearances and applications ... . This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers and provides invaluable reading for amateurs and professionals alike." (L'Enseignement Mathématique, Vol. 48 (1-2), 2002)

Spis treści

Foreword by Alena Solcová * Table of Contents * Preface * Glossary of Symbols * Introduction * Fundamentals of Number Theory * Basic Properties of Fermat Numbers * The Most Beautiful Theorems on Fermat Numbers * Primality of Fermat Numbers * Divisibility of Fermat Numbers * Factors of Fermat Numbers * Connection With the Pascal Triangle * Miscellaneous Results * The Irrationality of the Sum of Some Reciprocals * Fermat Primes and a Diophantine Equation * Fermat's Little Theorem, Pseudoprimes, and Super-Pseudoprimes * Generalizations of Fermat Numbers * Open Problems * Fermat Number Transform and Other Applications * The Proof of Gauss's Theorem * Euclidean Constructions of the Regular Heptadecagon * Appendix A, B, C * References * Web Site Sources * Name Index * Subject Index

Blurb

French mathematician Pierre de Fermat became most well known for his pioneering work in the area of number theory. His work with numbers has been attracting the attention of amateur and professional mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth and is based on a series of lectures given by the authors. The purpose of this book is to provide readers with an overview of the many properties of Fermat numbers and to demonstrate their numerous appearances and applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers and will provide invaluable reading for the amateur and professional alike.
Michal Krizek is a senior researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic and Associate Professor in the Department of Mathematics and Physics at Charles University in Prague. Florian Luca is a researcher at the Mathematical Institute of the UNAM in Morelia, Mexico. Lawrence Somer is a Professor of Mathematics at The Catholic University of America in Washington, D. C.