**Proofs from THE BOOK 6th Edition by Martin Aigner, ISBN-13: 978-3662572641**

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- Publisher: Springer; 6th ed. 2018 edition (July 6, 2018)
- Language: English
- 334 pages
- ISBN-10: 3662572648
- ISBN-13: 978-3662572641

This revised and enlarged sixth edition of * Proofs from THE BOOK *features an entirely new chapter on Van der Waerden’s permanent conjecture, as well as additional, highly original and delightful proofs in other chapters.

From the citation on the occasion of the 2018 “Steele Prize for Mathematical Exposition”

“… It is almost impossible to write a * mathematics* book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. […] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty.”

**Table of Contents:**

Preface

Preface to the Sixth Edition

Table of Contents

Number Theory

Chapter 1 Six proofs of the infinity of primes

Appendix: Infinitely many more proofs

References

Chapter 2 Bertrand’s postulate

Appendix: Some estimates

Estimating via integrals

Estimating factorials — Stirling’s formula

Estimating binomial coefficients

References

Chapter 3 Binomial coefficients are (almost) never powers

References

Chapter 4 Representing numbers as sums of two squares

References

Chapter 5 The law of quadratic reciprocity

References

Chapter 6 Every finite division ring is a field

References

Chapter 7 The spectral theorem and Hadamard’s determinant problem

References

Chapter 8 Some irrational numbers

References

Chapter 9 Four times π2/6

Appendix: The Riemann zeta function

References

**Geometry**

Chapter 10 Hilbert’s third problem: decomposing polyhedra

Appendix: Polytopes and polyhedra

References

Chapter 11 Lines in the plane and decompositions of graphs

Appendix: Basic graph concepts

References

Chapter 12 The slope problem

References

Chapter 13 Three applications of Euler’s formula

1. The Sylvester–Gallai theorem, revisited

2. Monochromatic lines

3. Pick’s theorem

References

Chapter 14 Cauchy’s rigidity theorem

References

Chapter 15 The Borromean rings don’t exist

Appendix: Basic notions on knots and links

References

Chapter 16 Touching simplices

References

Chapter 17 Every large point set has an obtuse angle

Appendix: Three tools from probability

References

Chapter 18 Borsuk’s conjecture

References

Analysis

Chapter 19 Sets, functions, and the continuum hypothesis

Appendix: On cardinal and ordinal numbers

References

Chapter 20 In praise of inequalities

References

Chapter 21 The fundamental theorem of **algebra**

References

Chapter 22 One square and an odd number of triangles

Appendix: Extending valuations

References

Chapter 23 A theorem of Pólya on polynomials

Appendix: Chebyshev’s theorem

References

Chapter 24 Van der Waerden’s permanent conjecture

References

Chapter 25 On a lemma of Littlewood and Offord

References

Chapter 26 Cotangent and the Herglotz trick

References

Chapter 27 Buffon’s needle problem

References

Combinatorics

Chapter 28 Pigeon-hole and double counting

1. Numbers

2. Sequences

3. Sums

4. Numbers again

5. Graphs

6. Sperner’s Lemma

References

Chapter 29 Tiling rectangles

References

Chapter 30 Three famous theorems on finite sets

References

Chapter 31 Shuffling cards

References

Chapter 32 Lattice paths and determinants

References

Chapter 33 Cayley’s formula for the number of trees

References

Chapter 34 Identities versus bijections

References

Chapter 35 The finite Kakeya problem

References

Chapter 36 Completing Latin squares

References

Graph Theory

Chapter 37 Permanents and the power of entropy

Appendix: More about entropy

References

Chapter 38 The Dinitz problem

References

Chapter 39 Five-coloring plane graphs

References

Chapter 40 How to guard a museum

References

Chapter 41 Turán’s graph theorem

References

Chapter 42 Communicating without errors

References

Chapter 43 The chromatic number of Kneser graphs

Appendix: A proof sketch for the Borsuk–Ulam theorem

References

Chapter 44 Of friends and politicians

References

Chapter 45 Probability makes counting (sometimes) easy

References

About the Illustrations

Index

* Martin Aigner* received his Ph.D. from the University of Vienna and has been professor of mathematics at the Freie Universität Berlin since 1974. He has published in various fields of combinatorics and graph theory and is the author of several monographs on discrete mathematics, among them the Springer books Combinatorial Theory and A Course on Enumeration. Martin Aigner is a recipient of the 1996 Lester R. Ford Award for mathematical exposition of the Mathematical Association of America MAA.

* Günter M. Ziegler *received his Ph.D. from

**M.I.T.**and has been professor of mathematics in Berlin – first at TU Berlin, now at Freie Universität – since 1995. He has published in discrete mathematics, geometry, topology, and optimization, including the Lectures on Polytopes with Springer, as well as „Do I Count? Stories from Mathematics“. Günter M. Ziegler is a recipient of the 2006 Chauvenet Prize of the MAA for his expository writing and the 2008 Communicator award of the German Science Foundation.

Martin Aigner and Günter M. Ziegler have started their work on Proofs from THE BOOK in 1995 together with Paul Erdös. The first edition of this book appeared in 1998 – it has since been translated into 13 languages: Brazilian, Chinese, German, Farsi, French, Hungarian, Italian, Japanese, Korean, Polish, Russian, Spanish, and Turkish.

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