Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."
Proofs from THE BOOK contains 32 sections (45 in the sixth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, analysis, combinatorics and graph theory. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by Karl Heinrich Hofmann [de]. It has gone through six editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian and Spanish.
In November 2017 the American Mathematical Society announced the 2018 Leroy P. Steele Prize for Mathematical Exposition to be awarded to Aigner and Ziegler for this book.
The proofs include:
Six proofs of the infinitude of the primes, including Euclid's and Furstenberg's
Proof of Bertrand's postulate
Fermat's theorem on sums of two squares
Two proofs of the Law of quadratic reciprocity
Proof of Wedderburn's little theorem asserting that every finite division ring is a field
Four proofs of the Basel problem
Proof that e is irrational (also showing the irrationality of certain related numbers)
Sylvester–Gallai theorem and De Bruijn–Erdős theorem
Cauchy's theorem
Borsuk's conjecture
Schröder–Bernstein theorem
Wetzel's problem on families of analytic functions with few distinct values
The fundamental theorem of algebra
Monsky's theorem (4th edition)
Van der Waerden's conjecture
Littlewood–Offord lemma
Buffon's needle problem
Sperner's theorem, Erdős–Ko–Rado theorem and Hall's theorem
Lindström–Gessel–Viennot lemma and the Cauchy–Binet formula
Four proofs of Cayley's formula
Kakeya sets in vector spaces over finite fields
Bregman–Minc inequality
Dinitz problem
Steve Fisk's proof of the art gallery theorem
Five proofs of Turán's theorem
Shannon capacity and Lovász number
Chromatic number of Kneser graphs
Friendship theorem
Some proofs using the Probabilistic method
References
Proofs from THE BOOK, Springer, Berlin 1998, ISBN 3-540-63698-6
Aigner, Martin; Ziegler, Günter (2009). Proofs from THE BOOK (4th ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-642-00855-9.
Table of Contents of the 4th ed.
Günter M. Ziegler's homepage, including a list of editions and translations.
Shepard, Mary (1999). "Read This! Review of Proofs from THE BOOK". Mathematical Association of America. .
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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