Fermat's Last Theorem for Amateurs

This book is intended for amateurs, students and teachers. The author presents partial results which could be obtained with exclusively elementary methods. The proofs are given in detail, with minimal prerequisites. An original feature are the ten interludes, devoted to important topics of elementary number theory, thus making the reading of this book self-contained. Their interest goes beyond Fermat's theorem. The Epilogue is a serious attempt to render accessible the strategy of the recent proof of Fermat's last theorem, a great mathematical feat.

Preliminary Booksellers Text: Do Not Use. In 1995, Andrew Wiles published two papers containing a proof of Fermat's Last Theorem. BRAVO FOR THIS GREAT MATHEMATICAL FEAT! Nevertheless, one shouldn't dismiss the earlier attempts to solve the problems. From giants in mathematics to clever amateurs, all did their best. In this book, aimed at amateurs, teachers, and mathematicians curious about the unfolding of the subject, the author restricts his attention exclusively to elementary methods. There are other books about Wiles' proof but the reader without an extended solid background may prefer to stay with this book.

Ribenboim's presentation of attempts to solve Fermat's Last Theorem sheds new light on the possibilities and power of the language of mathematics This book is intended for amateurs. All results presented were obtained with exclusively elementary methods; proofs are given in detail with minimal prerequisites Original features are the ten interludes, devoted to important topics of elementary number theory, thus making this book self-contained The Epilogue is a serious attempt to render accessible the strategy of Wiles recent proof of Fermat's Last Theorem, a great mathematical feat

Autorentext
Paulo Ribenboim ist emeritierter Professor der kanadischen Queen's University, Fellow der Royal Society of Canada und Träger des George Pólya-Preises der Mathematical Association of America. Er ist Autor von 13 Büchern und über 150 Forschungsartikeln.

Klappentext

ItisnowwellknownthatFermat slasttheoremhasbeenproved. For more than three and a half centuries, mathematicians from the greatnamestothecleveramateurs triedtoproveFermat sfamous statement. The approach was new and involved very sophisticated theories. Finallythelong-soughtproofwasachieved. Thearithmetic theory of elliptic curves, modular forms, Galois representations, and their deformations, developed by many mathematicians, were the tools required to complete the di?cult proof. Linked with this great mathematical feat are the names of TANI- YAMA, SHIMURA, FREY, SERRE, RIBET, WILES, TAYLOR. Their contributions, as well as hints of the proof, are discussed in the Epilogue. This book has not been written with the purpose of presentingtheproofofFermat stheorem. Onthecontrary, itiswr- ten for amateurs, teachers, and mathematicians curious about the unfolding of the subject. I employ exclusively elementary methods (except in the Epilogue). They have only led to partial solutions but their interest goes beyond Fermat s problem. One cannot stop admiring the results obtained with these limited techniques. Nevertheless, I warn that as far as I can see which in fact is not much the methods presented here will not lead to a proof of Fermat s last theorem for all exponents. vi Preface The presentation is self-contained and details are not spared, so the reading should be smooth. Most of the considerations involve ordinary rational numbers and only occasionally some algebraic (non-rational) numbers. For this reason I excluded Kummer s important contributions, which are treated in detail in my book, Classical Theory of Algebraic N- bers and described in my 13 Lectures on Fermat s Last Theorem (new printing, containing an Epilogue about recent results).

Zusammenfassung
Although this was certainly a great mathematical feat, one shouldn't dismiss earlier attempts made by mathematicians and clever amateurs to solve the problem. In this book, aimed at amateurs curious about the history of the subject, the author restricts his attention exclusively to elementary methods that have produced rich results.

Inhalt
The Problem.- Special Cases.- 4 Interludes.- Algebraic Restrictions on Hypothetical Solutions.- Germain's Theorem.- Interludes 5 and 6.- Arithmetic Restrictions on Hypothetical Solutions and on the Exponent.- Interludes 7 and 8.- Reformulations, Consequences, and Criteria.- Interludes 9 and 10.- The Local and Modular Fermat Problem.- Epilogue.