''Proofs from THE BOOK'' is a book of

mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...

s by

Martin Aigner Martin Aigner (born February 28, 1942 in Linz) is an Austrian mathematician and professor at Freie Universität Berlin since 1974 with interests in combinatorial mathematics and graph theory. He received his Ph.D from the University of Vienna. Hi ...

and

Günter M. Ziegler Günter Matthias Ziegler (born 19 May 1963) is a German mathematician who has been serving as president of the Free University of Berlin since 2018. Ziegler is known for his research in discrete mathematics and geometry, and particularly on the ...

. The book is dedicated to the

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Paul Erdős, who often referred to "The Book" in which

God In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...

keeps the most elegant proof of each mathematical

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."

Content

''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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

, combinatorics and

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by . 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, including

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

's and Furstenberg's *

Proof of Bertrand's postulate In mathematics, Bertrand's postulate (actually a theorem) states that for each n \ge 2 there is a prime p such that n. It was first

Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...

* Two proofs of the

Law of quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

* Proof of

Wedderburn's little theorem In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to al ...

asserting that every finite division ring is a field * Four proofs of the

Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...

Proof that e is irrational The e (mathematical constant), number ''e'' was introduced by Jacob Bernoulli in 1683. More than half a century later, Leonhard Euler, Euler, who had been a student of Jacob's younger brother Johann Bernoulli, Johann, proved that ''e'' is Irratio ...

(also showing the irrationality of certain related numbers) *

Hilbert's third problem The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...

Sylvester–Gallai theorem The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, ...

and De Bruijn–Erdős theorem * Cauchy's theorem * Borsuk's conjecture *

Schröder–Bernstein theorem In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if ...

* 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 In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function ...

* Littlewood–Offord lemma * Buffon's needle problem * Sperner's theorem, Erdős–Ko–Rado theorem and Hall's theorem *

Lindström–Gessel–Viennot lemma In Mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous w ...

and the

Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so tha ...

* Four proofs of

Cayley's formula In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the number of spanning tr ...

* Kakeya sets in vector spaces over finite fields *

Bregman–Minc inequality In discrete mathematics, the Bregman–Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Le ...

* Dinitz problem * Steve Fisk's proof of the

art gallery theorem Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas. There is no generally agreed definition of what ...

* Five proofs of

Turán's theorem In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the large ...

* Shannon capacity and

Lovász number In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász theta function and is commonly denoted by \vartheta(G), using a script form of the Greek letter ...

Chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...

of Kneser graphs * Friendship theorem * Some proofs using the

probabilistic method The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects fr ...

References

* **
Günter M. Ziegler's homepage
including a list of editions and translations. *

External links

Proofs from THE BOOK
(4th ed.) on the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...

Mathematical proofs Mathematics books Paul Erdős 1998 non-fiction books {{mathematics-lit-stub