Fermat’s Last Theorem for Amateurs

1. Front Matter

   Pages i-xiii

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2. The Problem

   Pages 1-2

3. Special Cases

   Pages 3-72

4. 4 Interludes

   Pages 73-98

5. Algebraic Restrictions on Hypothetical Solutions

   Pages 99-107

6. Germain’s Theorem

   Pages 109-142

7. Interludes 5 and 6

   Pages 143-163

8. Arithmetic Restrictions on Hypothetical Solutions and on the Exponent

   Pages 165-212

9. Interludes 7 and 8

   Pages 213-234

10. Reformulations, Consequences, and Criteria

    Pages 235-276

11. Interludes 9 and 10

    Pages 277-286

12. The Local and Modular Fermat Problem

    Pages 287-358

13. Epilogue

    Pages 359-380

14. Back Matter

    Pages 381-407

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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).

• Fermat
• Fermat's Last Theorem
• algebra
• arithmetic
• elliptic curve
• modular form
• number theory

From the reviews:

MATHEMATICAL REVIEWS

"The history of elementary approaches to Fermat is very rich indeed, and Ribenboim has arranged these approaches in a way that makes them accessible to interested readers without extensive mathematical backgrounds…both readable and fairly comprehensive. This book would likely be of great interest to an enthusiastic undergraduate with a basic knowledge of rings and fields. In addition to describing the history of one of the great problems in number theory, the book provides a gentle and well-motivated introduction to some important ideas in modern number theory…any reader who spends a few hours with this book is guaranteed to learn something new and interesting about Fermat’s last theorem."

• Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

  Paulo Ribenboim

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