Proofs from THE BOOK PDF
Preview Proofs from THE BOOK
Martin Aigner · Günter M. Ziegler Proofs from THE BOOK Sixth Edition MartinAigner GünterM.Ziegler Proofs from THE BOOK SixthEdition Martin Aigner Günter M. Ziegler Proofs from THE BOOK Sixth Edition IncludingIllustrationsbyKarlH.Hofmann 123 Martin Aigner Günter M. Ziegler Institut für Mathematik Institut für Mathematik Freie Universität Berlin Freie Universität Berlin Berlin, Germany Berlin, Germany ISBN 978-3-662-57264-1 ISBN 978-3-662-57265-8 (eBook) https://doi.org/10.1007/978-3-662-57265-8 Library of Congress Control Number: 2018940433 © Springer-Verlag GmbH Germany, part of Springer Nature 1998, 2001, 2004, 2010, 2014, 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany Preface PaulErdo˝slikedtotalkaboutTheBook,inwhichGodmaintainstheperfect proofsformathematicaltheorems,followingthedictumofG.H.Hardythat thereisnopermanentplaceforuglymathematics. Erdo˝salsosaidthatyou need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdo˝s’ 85th birthday. With Paul’s unfortunate death inthesummerof1996,heisnotlistedasaco-author. Insteadthisbookis dedicatedtohismemory. PaulErdo˝s Wehavenodefinitionorcharacterizationofwhatconstitutesaprooffrom The Book: all we offer here is the examples that we have selected, hop- ingthatourreaderswillshareourenthusiasmaboutbrilliantideas, clever insights and wonderful observations. We also hope that our readers will enjoythisdespitetheimperfectionsofourexposition. Theselectionistoa greatextentinfluencedbyPaulErdo˝shimself.Alargenumberofthetopics weresuggestedbyhim,andmanyoftheproofstracedirectlybacktohim, or were initiated by his supreme insight in asking the right question or in makingtherightconjecture.Sotoalargeextentthisbookreflectstheviews ofPaulErdo˝sastowhatshouldbeconsideredaprooffromTheBook. “TheBook” Alimitingfactorforourselectionoftopicswasthateverythinginthisbook is supposed to be accessible to readers whose backgrounds include only a modest amount of technique from undergraduate mathematics. A little linearalgebra,somebasicanalysisandnumbertheory,andahealthydollop of elementary concepts and reasonings from discrete mathematics should besufficienttounderstandandenjoyeverythinginthisbook. We are extremely grateful to the many people who helped and supported us with this project — among them the students of a seminar where we discussedapreliminaryversion,toBennoArtmann,StephanBrandt,Stefan Felsner, Eli Goodman, Torsten Heldmann, and Hans Mielke. We thank Margrit Barrett, Christian Bressler, Ewgenij Gawrilow, Michael Joswig, Elke Pose, and Jörg Rambau for their technical help in composing this book. We are in great debt to Tom Trotter who read the manuscript from first to last page, to Karl H. Hofmann for his wonderful drawings, and mostofalltothelategreatPaulErdo˝shimself. Berlin,March1998 MartinAigner · GünterM.Ziegler VI Preface to the Sixth Edition The idea tothis project was born during some leisurelydiscussions at the MathematischesForschungsinstitutinOberwolfachwiththeincomparable Paul Erdo˝s in the mid-1990s. It is now nearly twenty years ago that we presented the first edition of our book on occasion of the International Congress of Mathematicians in Berlin 1998. At that time we could not possibly imagine the wonderful and lasting response our book about The Bookwouldhave,withallthewarmletters,interestingcommentsandsug- gestions,neweditions,andasofnowthirteentranslations. Itisnoexagger- ationtosaythatithasbecomeapartofourlives. Inadditiontonumerousimprovementsandsmallerchanges,manyofthem suggestedbyourreaders,forthepresentsixtheditionwewroteanentirely newchapterwithGurvits’sproofofVanderWaerden’spermanentconjec- ture,usedthistoderiveasymptoticsforthe(cid:2)numberofLatinsquares,added a new, fourth proof for the Euler theorem 1 = π2/6, and present n≥1 n2 a new geometric explanation for Heath-Brown’s involution proof for the Fermattwosquarestheorem. Wethankeveryonewhohelpedandencouragedusoveralltheseyears. For thesecondeditionthisincludedStephanBrandt,ChristianElsholtz,Jürgen Elstrodt, Daniel Grieser, Roger Heath-Brown, Lee L. Keener, Christian Lebœuf, HanfriedLenz, Nicolas Puech, JohnScholes,Bernulf Weißbach, and many others. The third edition benefitted especially from input by David Bevan, Anders Björner, Dietrich Braess, John Cosgrave, Hubert Kalf, Günter Pickert, Alistair Sinclair, and Herb Wilf. For the fourth edi- tion,wewereparticularlyindebtedtoOliverDeiser,AntonDochtermann, Michael Harbeck, Stefan Hougardy, Hendrik W. Lenstra, Günter Rote, MoritzW.Schmitt,andCarstenSchultzfortheircontributions.Forthefifth edition, we gratefully acknowledged ideas and suggestions by Ian Agol, France Dacar, Christopher Deninger, Michael D. Hirschhorn, Franz Lemmermeyer, Raimund Seidel, Tord Sjödin, and John M. Sullivan, as wellashelpfromMarie-SophieLitz,MiriamSchlöter,andJanSchneider. Forthepresentsixthedition,veryvaluablehintswereprovidedbyFrance Dacar again, as well as by David Benko, Jan Peter Schäfermeyer, and YuliyaSemikina. Moreover,wethankRuthAlleweltatSpringerinHeidelbergandChristoph Eyrich,TorstenHeldmann,andElkePoseinBerlinfortheircontinuingsup- portthroughouttheseyears.Andfinally,thisbookwouldcertainlynotlook thesamewithouttheoriginaldesignsuggestedbyKarl-FriedrichKoch,and thesuperbnewdrawingsprovidedagainandagainbyKarlH.Hofmann. Berlin,March2018 MartinAigner · GünterM.Ziegler Table of Contents Number Theory 1 1. Sixproofsoftheinfinityofprimes .............................. 3 2. Bertrand’spostulate ........................................... 9 3. Binomialcoefficientsare(almost)neverpowers ................. 15 4. Representingnumbersassumsoftwosquares .................. 19 5. Thelawofquadraticreciprocity ............................... 27 6. Everyfinitedivisionringisafield ............................. 35 7. ThespectraltheoremandHadamard’sdeterminantproblem ...... 39 8. Someirrationalnumbers ...................................... 47 9. Fourtimesπ2/6 ............................................. 55 Geometry 65 10. Hilbert’sthirdproblem: decomposingpolyhedra ................ 67 11. Linesintheplaneanddecompositionsofgraphs ................ 77 12. Theslopeproblem ........................................... 83 13. ThreeapplicationsofEuler’sformula .......................... 89 14. Cauchy’srigiditytheorem .................................... 95 15. TheBorromeanringsdon’texist .............................. 99 16. Touchingsimplices ......................................... 107 17. Everylargepointsethasanobtuseangle ...................... 111 18. Borsuk’sconjecture ......................................... 117 Analysis 125 19. Sets,functions,andthecontinuumhypothesis ................. 127 20. Inpraiseofinequalities ..................................... 143 21. Thefundamentaltheoremofalgebra .......................... 151 22. Onesquareandanoddnumberoftriangles .................... 155 VIII TableofContents 23. AtheoremofPólyaonpolynomials .......................... 163 24. VanderWaerden’spermanentconjecture ...................... 169 25. OnalemmaofLittlewoodandOfford ........................ 179 26. CotangentandtheHerglotztrick ............................. 183 27. Buffon’sneedleproblem .................................... 189 Combinatorics 193 28. Pigeon-holeanddoublecounting ............................. 195 29. Tilingrectangles ............................................ 207 30. Threefamoustheoremsonfinitesets ......................... 213 31. Shufflingcards ............................................. 219 32. Latticepathsanddeterminants ............................... 229 33. Cayley’sformulaforthenumberoftrees ...................... 235 34. Identitiesversusbijections ................................... 241 35. ThefiniteKakeyaproblem ................................... 247 36. CompletingLatinsquares ................................... 253 Graph Theory 259 37. Permanentsandthepowerofentropy ......................... 261 38. TheDinitzproblem ......................................... 271 39. Five-coloringplanegraphs .................................. 277 40. Howtoguardamuseum ..................................... 281 41. Turán’sgraphtheorem ...................................... 285 42. Communicatingwithouterrors ............................... 291 43. ThechromaticnumberofKnesergraphs ...................... 301 44. Offriendsandpoliticians .................................... 307 45. Probabilitymakescounting(sometimes)easy ................. 311 About the Illustrations 321 Index 323 Number Theory 1 Sixproofs oftheinfinityofprimes 3 2 Bertrand’spostulate 9 3 Binomialcoefficients are(almost)neverpowers 15 4 Representingnumbers assumsoftwosquares 19 5 Thelawof quadraticreciprocity 27 6 Everyfinitedivisionring isafield 35 7 Thespectraltheorem andHadamard’s determinantproblem 39 8 Someirrationalnumbers 47 9 Fourtimesπ2/6 55 “Irrationalityandπ” Six proofs Chapter 1 of the infinity of primes It is only natural that we start these notes with probably the oldest Book Proof, usually attributed to Euclid (Elements IX, 20). It shows that the sequenceofprimesdoesnotend. (cid:2) Euclid’s Proof. For any finite set {p ,...,p } of primes, consider 1 r the number n = p p ···p +1. This n has a prime divisor p. But p is 1 2 r not one of the p : otherwise p would be a divisor of n and of the product i p p ···p , and thus also of the difference n−p p ···p = 1, which is 1 2 r 1 2 r impossible.Soafiniteset{p ,...,p }cannotbethecollectionofallprime 1 r numbers. (cid:3) Before we continue let us fix some notation. N = {1,2,3,...} is the set ofnaturalnumbers,Z = {...,−2,−1,0,1,2,...}thesetofintegers,and P={2,3,5,7,...}thesetofprimes. Inthefollowing,wewillexhibitvariousotherproofs(outofamuchlonger list)whichwehopethereaderwilllikeasmuchaswedo. Althoughthey usedifferentview-points,thefollowingbasicideaiscommontoallofthem: The natural numbers grow beyond all bounds, and every natural number n ≥ 2 has a prime divisor. These two facts taken together force P to be infinite. ThenextproofisduetoChristianGoldbach(fromalettertoLeon- hard Euler 1730), the third proof is apparently folklore, the fourth one is byEulerhimself,thefifthproofwasproposedbyHarryFürstenberg,while thelastproofisduetoPaulErdo˝s. (cid:2)SecondProof. LetusfirstlookattheFermatnumbersF =22n+1for F0 = 3 n F1 = 5 n = 0,1,2,.... WewillshowthatanytwoFermatnumbersarerelatively F2 = 17 prime;hencetheremustbeinfinitelymanyprimes. Tothisend,weverify F3 = 257 therecursion n(cid:3)−1 F4 = 65537 Fk = Fn−2 (n≥1), F5 = 641·6700417 k=0 ThefirstfewFermatnumbers fromwhichourassertionfollowsimmediately. Indeed,ifmisadivisorof, say, F and F (k < n), then m divides 2, and hence m = 1 or 2. But k n m=2isimpossiblesinceallFermatnumbersareodd. Toprovetherecursionweuseinductiononn. Forn = 1wehaveF = 3 0 andF −2=3. Withinductionwenowconclude 1 (cid:3)n (cid:4)n(cid:3)−1 (cid:5) F = F F = (F −2)F = k k n n n k=0 k=0 =(22n −1)(22n +1) = 22n+1 −1 = F −2. (cid:3) n+1 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 M. Aigner, G. M. Ziegler, Proofs from THE BOOK, https://doi.org/10.1007/978-3-662-57265-8_1
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