Proofs from THE BOOK

1. Front Matter

   Pages I-VIII

   PDF

2. Number Theory

   1. Front Matter

      Pages 1-1

      PDF

   2. Six proofs of the infinity of primes

      • Martin Aigner, Günter M. Ziegler

      Pages 3-6

   3. Bertrand’s postulate

      • Martin Aigner, Günter M. Ziegler

      Pages 7-12

   4. Binomial coefficients are (almost) never powers

      • Martin Aigner, Günter M. Ziegler

      Pages 13-16

   5. Representing numbers as sums of two squares

      • Martin Aigner, Günter M. Ziegler

      Pages 17-22

   6. The law of quadratic reciprocity

      • Martin Aigner, Günter M. Ziegler

      Pages 23-30

   7. Every finite division ring is a field

      • Martin Aigner, Günter M. Ziegler

      Pages 31-34

   8. Some irrational numbers

      • Martin Aigner, Günter M. Ziegler

      Pages 35-41

   9. Three times Π²/6

      • Martin Aigner, Günter M. Ziegler

      Pages 43-50

3. Geometry

   1. Front Matter

      Pages 51-51

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   2. Hilbert’s third problem: decomposing polyhedra

      • Martin Aigner, Günter M. Ziegler

      Pages 53-61

   3. Lines in the plane and decompositions of graphs

      • Martin Aigner, Günter M. Ziegler

      Pages 63-67

   4. The slope problem

      • Martin Aigner, Günter M. Ziegler

      Pages 69-73

   5. Three applications of Euler’s formula

      • Martin Aigner, Günter M. Ziegler

      Pages 75-80

   6. Cauchy’s rigidity theorem

      • Martin Aigner, Günter M. Ziegler

      Pages 81-84

   7. Touching simplices

      • Martin Aigner, Günter M. Ziegler

      Pages 85-88

   8. Every large point set has an obtuse angle

      • Martin Aigner, Günter M. Ziegler

      Pages 89-94

   9. Borsuk’s conjecture

      • Martin Aigner, Günter M. Ziegler

      Pages 95-100

4. Analysis

   1. Front Matter

      Pages 101-101

      PDF

PaulErdos ? likedtotalkaboutTheBook,inwhichGodmaintainstheperfect proofsformathematicaltheorems,followingthedictumofG. H. Hardythat there is no permanent place for ugly mathematics. Erdos ? also said that you 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 ?rst (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, ?lling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdos ? ’ 85th birthday. With Paul’s unfortunate death in the summer of 1996, he is not listed as a co-author. Instead this book is dedicated to his memory. ? Paul Erdos We have no de?nition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, h- ing that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations. We also hope that our readers will enjoy this despite the imperfections of our exposition. The selection is to a ? great extent in?uencedby Paul Erdos himself. A largenumberof the topics were suggested by him, and many of the proofs trace directly back to him, or were initiated by his supreme insight in asking the right question or in makingthe rightconjecture. So to a largeextentthisbookre?ectstheviews of Paul Erdos ? as to what should be considered a proof from The Book.

• Algebra
• Counting
• Finite
• Identity
• analysis
• calculus
• combinatorics
• function
• geometry
• number theory
• proof
• proofs
• theorem

From the reviews of the fourth edition:

“This is the fourth edition of a book that became a classic on its first appearance in 1998. … The authors have tried, in homage to Erdős, to approximate this tome; successive editions appear to be achieving uniform convergence. … Five new chapters have been added … . there is enough new material that libraries certainly should do so. For individuals who do not yet have their own copies, the argument for purchase has just grown stronger.” (Robert Dawson, Zentralblatt MATH, February, 2010)

“This book is the fourth edition of Aigner and Ziegler’s attempt to find proofs that Erdos would find appealing. … this one is a great collection of remarkable results with really nice proofs. The authors have done an excellent job choosing topics and proofs that Erdos would have appreciated. … the proofs are largely accessible to readers with an undergraduate-level mathematics background. … I love the fact that the chapters are relatively short and self-contained. … this is a very nice book.” (Donald L. Vestal, The Mathematical Association of America, May, 2010)

“Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdős. The theorems are so fundamental, their proofs so elegant, and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. The book has five parts of roughly equal length.” (Miklόs Bόna, The Book Review Column, 2011)

“Paul Erdős … had his own way of judging the beauty of various proofs. He said that there was a book somewhere, possibly in heaven, and that book contained the nicest and most elucidating proof of every theorem in mathematics. … Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems … that would undoubtedly be in the Book of Erdős. The theorems are so fundamental … that every mathematician, regardless of speciality, can benefit from reading this book.” (Miklós Bóna, SIGACT News, Vol. 42. (3), September, 2011)

From the reviews of the third edition:

"... It is unusual for a reviewer to have the opportunity to review the first three editions of a book - the first edition was published in 1998, the second in 2001 and the third in 2004. ... I was fortunate enough to obtain a copy of the first edition while travelling in Europe in 1999 and I spent many pleasant hours reading it carefully from cover to cover. The style is inviting and it is very hard to stop part way through a chapter. Indeed I have recommended the book to talented undergraduates and to mathematically literate friends. All report that they are captivated by the material and the new view of mathematics it engenders. By now a number of reviews of the earlier editions have appeared and I must simply agree that the book is a pleasure to hold and to look at, it has striking photographs, instructive pictures and beautiful drawings. The style is clear and entertaining and the proofs are brilliant and memorable. ...

David Hunt, The Mathematical Gazette, Vol. 32, Issue 2, p. 127-128

"The newest edition contains three completely new chapters. … The approach is refreshingly straightforward, all the necessary results from analysis being summarised in boxes, and a short appendix discusses the importance of the zeta-function in number theory. … this edition also contains additional material interpolated in the original text, notably the Calkin-Wilf enumeration of the rationals." (Gerry Leversha, The Mathematical Gazette, March, 2005)

"A lot of solid mathematics is packed into Proofs. Its thirty chapters, divided into sections on Number Theory, Geometry, Analysis … . Each chapter is largely independent; some include necessary background as an appendix. … The key to the approachability of Proofs lies not so much in the accessibility of its mathematics, however, as in the rewards it offers: elegant proofs of interesting results, which don’t leave the reader feeling cheated or disappointed." (Zentralblatt für Didaktik de Mathematik, July, 2004)

• FU Berlin, Berlin, Germany

  Martin Aigner, Günter M. Ziegler

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