Fermat's last theorem for amateurs

423 Pages·1999·2.22 MB·English

by  Ribenboim P.

Paulo Ribenboim
Department of Mathematics
and Statistics
Queen's University
Kingston, Ontario, K7L 3N6
Canada
With 2 Illustrations
Mathematics Subject Classification (1991): l!Axx
Library of Congress Cataloging-in-Publication Data
Ribenboim, Paulo.
Fermat's last theorem for amateurs l Paulo Ribenboim.
p. cm.
Includes bibliographical references and index.
ISBN 0-387-98508-5 (hc. : alk. paper)
l. Fermat's last theorem. l. Title.
QA244.R53 1999
512'.74---dc21 98-41246
© 1999 Springer-Verlag New York, !ne.
Ali rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag Ne w York, In c., 175 Fifth Avenue, Ne w Yo rk,
NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of generai descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 0-387-98508-5 Springer-Verlag New York Berlin Heidelberg SPIN 10753540
Preface
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 difficult 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,itiswrit-
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 Num-
bers and described in my 13 Lectures on Fermat’s Last Theorem
(new printing, containing an Epilogue about recent results).
There are already — and there will be more — books, mono-
graphs, and papers explaining the ideas and steps in the proof of
Fermat’s theorem. The readers with an extended solid background
will profit more from reading such writings. Others may prefer to
stay with me.
In summary, if you are an amateur or a young beginner, you may
love what you will read here, as I made a serious effort to provide
thorough and clear explanations.
On the other hand, if you are a professional mathematician, you
may then wonder why I have undertaken this task now that the
problem has been solved. The tower of Babel did not reach the
sky, but it was one of the marvels of ancient times. Here too, there
are some admirable examples of ingenuity, even more remarkable
considering that the arguments are strictly elementary. It would be
an unforgivable error to let these gems sink into oblivion. As Jacobi
said, all for “l’honneur de l’esprit humain.”
August, 1997 Paulo Ribenboim
Reader
You may feel tempted to write your own (simpler) proof of Fermat’s
last theorem.
I have strong views about such a project. It should be written in
the Constitution of States and Nations, in the Chapter of Human
Rights:
It is an inalienable right of each individual to produce his or her
own proof of Fermat’s last theorem.
However, such a solemn statement about Fermat’s last theorem
(henceforth referred to as THE theorem) should be tempered by the
following articles:
Art. 1. No attempted proof of THE theorem should
ever duplicate a previous one.
Art. 2. It is a criminal offense to submit false proofs
of THE theorem to professors who arduously earn
theirlivingbyteachinghownottoconceivefalseproofs
of THE theorem.
Infringement of the latter, leads directly to Hell. Return to Par-
adise only after the said criminal has understood and is able to re-
produce Wiles’ proof. (Harsh punishment.)
This page intentionally left blank
Contents
Preface v
Reader vii
Acknowledgment xiii
The Problem 1
I. Special Cases 3
I.1. The Pythagorean Equation 3
I.2. The Biquadratic Equation 11
I.3. Gaussian Numbers 21
I.4. The Cubic Equation 24
I.5. The Eisenstein Field 41
I.6. The Quintic Equation 49
I.7. Fermat’s Equation of Degree Seven 57
I.8. Other Special Cases 63
I.9. Appendix 71