《计算机问题求解》课程教学资源（参考书籍）Proofs from THE BOOK（Fifth Edition，Martin Aigner · Günter M. Ziegler）

Martin Aigner.Gunter M.Ziegler Proofs from THE BOOK Fifth Edition ②Springer

Martin Aigner · Günter M. Ziegler Proofs from THE BOOK Fifth Edition

Preface Paul Erdos liked to talk about The Book,in which God maintains the perfect proofs for mathematical theorems,following the dictum of G.H.Hardy that 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 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 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 Erd6s We have no definition or characterization of what constitutes a proof from The Book:all we offer here is the examples that we have selected,hop- 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 influenced by Paul Erdos himself.A large number of 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 making the right conjecture.So to a large extent this book reflects the views of Paul Erdos as to what should be considered a proof from The Book “The Book A limiting factor for our selection of topics was that everything in this book is supposed to be accessible to readers whose backgrounds include only a modest amount of technique from undergraduate mathematics.A little linear algebra,some basic analysis and number theory,and a healthy dollop of elementary concepts and reasonings from discrete mathematics should be sufficient to understand and enjoy everything in this book. 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 discussed a preliminary version,to Benno Artmann,Stephan Brandt,Stefan Felsner,Eli Goodman,Torsten Heldmann,and Hans Mielke.We thank Margrit Barrett,Christian Bressler,Ewgenij Gawrilow,Michael Joswig, Elke Pose,and Jorg 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 most of all to the late great Paul Erdos himself. Berlin,March 1998 Martin Aigner.Giinter M.Ziegler

Preface Paul Erdos˝ Paul Erdos liked to talk about The Book, in which God maintains the perfect ˝ proofs for mathematical theorems, following the dictum of G. H. Hardy that 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 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 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. “The Book” We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hop￾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 influenced by Paul Erdos himself. A large number of 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 making the right conjecture. So to a large extent this book reflects the views of Paul Erdos as to what should be considered a proof from The Book. ˝ A limiting factor for our selection of topics was that everything in this book is supposed to be accessible to readers whose backgrounds include only a modest amount of technique from undergraduate mathematics. A little linear algebra, some basic analysis and number theory, and a healthy dollop of elementary concepts and reasonings from discrete mathematics should be sufficient to understand and enjoy everything in this book. 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 discussed a preliminary version, to Benno Artmann, Stephan Brandt, 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 most of all to the late great Paul Erdos himself. ˝ Berlin, March 1998 Martin Aigner · Günter M. Ziegler

VI Preface to the Fifth Edition It is now almost twenty years that the idea to this project was born dur- ing some leisurely discussions at the Mathematisches Forschungsinstitut in Oberwolfach with the incomparable Paul Erdos.At that time we could not possibly imagine the wonderful and lasting response our book about The Book would have,with all the warm letters,interesting comments and suggestions,new editions,and as of now thirteen translations.It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements and smaller changes,many of them suggested by our readers,the present fifth edition contains four new chap- ters,which present an extraordinary proof for the classical spectral theorem from linear algebra,the impossibility of the Borromean rings as a highlight from geometry,the finite version of Kakeya's problem,and an inspired proof for Minc's permanent conjecture. We thank everyone who helped and encouraged us over all these years.For the second edition this included Stephan Brandt,Christian Elsholtz,Jurgen Elstrodt,Daniel Grieser,Roger Heath-Brown,Lee L.Keener,Christian Leboeuf,Hanfried Lenz,Nicolas Puech,John Scholes,Bernulf WeiBbach, and many others.The third edition benefitted especially from input by David Bevan,Anders Bjorner,Dietrich Braess,John Cosgrave,Hubert Kalf,Gunter Pickert,Alistair Sinclair,and Herb Wilf.For the fourth edi- tion,we were particularly indebted to Oliver Deiser,Anton Dochtermann, Michael Harbeck,Stefan Hougardy,Hendrik W.Lenstra,Guinter Rote, Moritz W.Schmitt,and Carsten Schultz for their contributions.For the present edition,we gratefully acknowledge ideas and suggestions by Ian Agol,France Dacar,Christopher Deninger,Michael D.Hirschhorn,Franz Lemmermeyer,Raimund Seidel,Tord Sjodin,and John M.Sullivan,as well as help from Marie-Sophie Litz,Miriam Schloter,and Jan Schneider. Moreover,we thank Ruth Allewelt at Springer in Heidelberg and Christoph Eyrich,Torsten Heldmann,and Elke Pose in Berlin for their continuing sup- port throughout these years.And finally,this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch,and the superb new drawings provided for each edition by Karl H.Hofmann. Berlin,June 2014 Martin Aigner.Giinter M.Ziegler

VI Preface to the Fifth Edition It is now almost twenty years that the idea to this project was born dur￾ing some leisurely discussions at the Mathematisches Forschungsinstitut in Oberwolfach with the incomparable Paul Erdos. At that time we could ˝ not possibly imagine the wonderful and lasting response our book about The Book would have, with all the warm letters, interesting comments and suggestions, new editions, and as of now thirteen translations. It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements and smaller changes, many of them suggested by our readers, the present fifth edition contains four new chap￾ters, which present an extraordinary proof for the classical spectral theorem from linear algebra, the impossibility of the Borromean rings as a highlight from geometry, the finite version of Kakeya’s problem, and an inspired proof for Minc’s permanent conjecture. We thank everyone who helped and encouraged us over all these years. For the second edition this included Stephan Brandt, Christian Elsholtz, Jürgen Elstrodt, Daniel Grieser, Roger Heath-Brown, Lee L. Keener, Christian Lebœuf, Hanfried Lenz, Nicolas Puech, John Scholes, 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, we were particularly indebted to Oliver Deiser, Anton Dochtermann, Michael Harbeck, Stefan Hougardy, Hendrik W. Lenstra, Günter Rote, Moritz W. Schmitt, and Carsten Schultz for their contributions. For the present edition, we gratefully acknowledge ideas and suggestions by Ian Agol, France Dacar, Christopher Deninger, Michael D. Hirschhorn, Franz Lemmermeyer, Raimund Seidel, Tord Sjödin, and John M. Sullivan, as well as help from Marie-Sophie Litz, Miriam Schlöter, and Jan Schneider. Moreover, we thank Ruth Allewelt at Springer in Heidelberg and Christoph Eyrich, Torsten Heldmann, and Elke Pose in Berlin for their continuing sup￾port throughout these years. And finally, this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch, and the superb new drawings provided for each edition by Karl H. Hofmann. Berlin, June 2014 Martin Aigner · Günter M. Ziegler

Table of Contents Number Theory_ .1 1.Six proofs of the infinity of primes..............................3 2.Bertrand'spostulate.......................................9 3.Binomial coefficients are(almost)never powers.................15 4.Representing numbers as sums of two squares...................19 5.The law of quadratic reciprocity ...............25 6.Every finite division ring is a field..............................33 7.The spectral theorem and Hadamard's determinant problem.......37 8.Some irrational numbers......................................45 9.Three times2/6................53 Geometry_ 61 10.Hilbert's third problem:decomposing polyhedra.................63 11.Lines in the plane and decompositions of graphs.................73 12.The slope problem................................79 13.Three applications of Euler's formula..........................85 14.Cauchy's rigidity theorem.....................................91 15.The Borromean rings don't exist...............................95 16.Touching simplices..........................................103 17.Every large point set has an obtuse angle...................... 107 18.Borsuk'sconjecture.........................................113 Analysis 121 19.Sets,functions,and the continuum hypothesis..................123 20.In praise of inequalities......................................139 21.The fundamental theorem of algebra..........................147 22.One square and an odd number of triangles ....................151

Table of Contents Number Theory 1 1. Six proofs of the infinity of primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Bertrand’s postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. Binomial coefficients are (almost) never powers . . . . . . . . . . . . . . . . . 15 4. Representing numbers as sums of two squares . . . . . . . . . . . . . . . . . . . 19 5. The law of quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6. Every finite division ring is a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7. The spectral theorem and Hadamard’s determinant problem . . . . . . .37 8. Some irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 9. Three times π2/6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Geometry 61 10. Hilbert’s third problem: decomposing polyhedra . . . . . . . . . . . . . . . . .63 11. Lines in the plane and decompositions of graphs . . . . . . . . . . . . . . . . . 73 12. The slope problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 13. Three applications of Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 85 14. Cauchy’s rigidity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 15. The Borromean rings don’t exist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 16. Touching simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 17. Every large point set has an obtuse angle . . . . . . . . . . . . . . . . . . . . . . 107 18. Borsuk’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Analysis 121 19. Sets, functions, and the continuum hypothesis . . . . . . . . . . . . . . . . . . 123 20. In praise of inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 21. The fundamental theorem of algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 147 22. One square and an odd number of triangles . . . . . . . . . . . . . . . . . . . . 151

VIⅢ Table of Contents 23.A theorem of Polya on polynomials ...........................159 24.On a lemma of Littlewood and Offord.........................165 25.Cotangent and the Herglotz trick..............................169 26.Buffon's needle problem.....................................175 Combinatorics 179 27.Pigeon-hole and double counting............................. 181 28.Tiling rectangles............................................ 193 29.Three famous theorems on finite sets..........................199 30.Shuffing cards.............................................205 31.Lattice paths and determinants................................215 32.Cayley's formula for the number of trees...................... 221 33.Identities versus bijections...................................227 34.The finite Kakeya problem...................................233 35.Completing Latin squares 239 Graph Theory 245 36.The Dinitz problem........................................247 37.Permanents and the power of entropy .253 38.Five-coloring plane graphs ................................ 261 39.How to guard a museum.........265 40.Turan's graph theorem...................................... .269 41.Communicating without errors...............................275 42.The chromatic number of Kneser graphs.......................285 43.Of friends and politicians....................................291 44.Probability makes counting(sometimes)easy..................295 About the lllustrations 304 Index 305

VIII Table of Contents 23. A theorem of Pólya on polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 24. On a lemma of Littlewood and Offord . . . . . . . . . . . . . . . . . . . . . . . . . 165 25. Cotangent and the Herglotz trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 26. Buffon’s needle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Combinatorics 179 27. Pigeon-hole and double counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 28. Tiling rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 29. Three famous theorems on finite sets . . . . . . . . . . . . . . . . . . . . . . . . . . 199 30. Shuffling cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 31. Lattice paths and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 32. Cayley’s formula for the number of trees . . . . . . . . . . . . . . . . . . . . . . 221 33. Identities versus bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 34. The finite Kakeya problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 35. Completing Latin squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Graph Theory 245 36. The Dinitz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247 37. Permanents and the power of entropy . . . . . . . . . . . . . . . . . . . . . . . . . .253 38. Five-coloring plane graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 39. How to guard a museum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 40. Turán’s graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 41. Communicating without errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 42. The chromatic number of Kneser graphs . . . . . . . . . . . . . . . . . . . . . . . 285 43. Of friends and politicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 44. Probability makes counting (sometimes) easy . . . . . . . . . . . . . . . . . . 295 About the Illustrations 304 Index 305

Number Theory 1 Six proofs of the infinity of primes 3 2 Bertrand's postulate 9 3 Binomial coefficients are(almost)never powers 15 4 Representing numbers as sums of two squares 19 5 The law of quadratic reciprocity 25 6 Every finite division ring is a field 33 7 The spectral theorem and Hadamard's determinant problem 37 8 Some irrational numbers 45 9 Three timesπ2/653 “Irrationality and

Number Theory 1 Six proofs of the infinity of primes 3 2 Bertrand’s postulate 9 3 Binomial coefficients are (almost) never powers 15 4 Representing numbers as sums of two squares 19 5 The law of quadratic reciprocity 25 6 Every finite division ring is a field 33 7 The spectral theorem and Hadamard’s determinant problem 37 8 Some irrational numbers 45 9 Three times π2/6 53 “Irrationality and π

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 sequence of primes does not end. Euclid's Proof.For any finite set {pi,...,pr}of primes,consider the number n pip2...pr+1.This n has a prime divisor p.But p is not one of the pi:otherwise p would be a divisor of n and of the product pip2...pr,and thus also of the difference n-pip2...p =1,which is impossible.So a finite set [p1,...,pr}cannot be the collection of all prime numbers. ▣ Before we continue let us fix some notation.N={1,2,3,...}is the set of natural numbers,=f...,-2,-1,0,1,2,...!the set of integers,and P=12,3,5,7,...}the set of primes. In the following,we will exhibit various other proofs (out of a much longer list)which we hope the reader will like as much as we do.Although they use different view-points,the following basic idea is common to all of them: 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.The next proof is due to Christian Goldbach(from a letter to Leon- hard Euler 1730),the third proof is apparently folklore,the fourth one is by Euler himself,the fifth proof was proposed by Harry Furstenberg,while the last proof is due to Paul Erdos. Second Proof.Let us first look at the Fermat numbers Fn=22"+1for Fo =3 F =5 n =0,1,2,....We will show that any two Fermat numbers are relatively F2 =17 prime;hence there must be infinitely many primes.To this end,we verify F =257 the recursion n-1 F4= 65537 R=-2(m≥1)， Fs =641.6700417 k=0 The first few Fermat numbers from which our assertion follows immediately.Indeed,if m is a divisor of, say,Fk and Fn (k<n),then m divides 2,and hence m 1 or 2.But m=2 is impossible since all Fermat numbers are odd. To prove the recursion we use induction on n.For n =1 we have Fo =3 and F1-2 =3.With induction we now conclude ⅡA=()=G-, n-l =(22"-1)(22"+1)=22m+-1=F+1-2. ▣ M.Aigner,G.M.Ziegler,Proofs from THE BOOK,DOI 10.1007/978-3-662-44205-0_1, @Springer-Verlag Berlin Heidelberg 2014

Six proofs of the infinity of primes Chapter 1 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 sequence of primes does not end. Euclid’s Proof. For any finite set {p1,...,pr} of primes, consider the number n = p1p2 ··· pr + 1. This n has a prime divisor p. But p is not one of the pi: otherwise p would be a divisor of n and of the product p1p2 ··· pr, and thus also of the difference n − p1p2 ··· pr = 1, which is impossible. So a finite set {p1,...,pr} cannot be the collection of all prime numbers. Before we continue let us fix some notation. N = {1, 2, 3,...} is the set of natural numbers, Z = {..., −2, −1, 0, 1, 2,...} the set of integers, and P = {2, 3, 5, 7,...} the set of primes. In the following, we will exhibit various other proofs (out of a much longer list) which we hope the reader will like as much as we do. Although they use different view-points, the following basic idea is common to all of them: 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. The next proof is due to Christian Goldbach (from a letter to Leon￾hard Euler 1730), the third proof is apparently folklore, the fourth one is by Euler himself, the fifth proof was proposed by Harry Fürstenberg, while the last proof is due to Paul Erdos. ˝ Second Proof. Let us first look at the Fermat numbers Fn = 22n +1 for n = 0, 1, 2,.... We will show that any two Fermat numbers are relatively prime; hence there must be infinitely many primes. To this end, we verify the recursion n −1 k=0 Fk = Fn − 2 (n ≥ 1), from which our assertion follows immediately. Indeed, if m is a divisor of, F0 = 3 F1 = 5 F2 = 17 F3 = 257 F4 = 65537 F5 = 641 · 6700417 The first few Fermat numbers say, Fk and Fn (k<n), then m divides 2, and hence m = 1 or 2. But m = 2 is impossible since all Fermat numbers are odd. To prove the recursion we use induction on n. For n = 1 we have F0 = 3 and F1 − 2=3. With induction we now conclude n k=0 Fk = n −1 k=0 Fk  Fn = (Fn − 2)Fn = = (22n − 1)(22n + 1) = 22n+1 − 1 = Fn+1 − 2. M. Aigner, G.M. Ziegler, Proofs from THE BOOK, DOI 10.1007/978-3-662-44205-0_1, © Springer-Verlag Berlin Heidelberg 2014

Six proofs of the infinity of primes Third Proof.Suppose P is finite and pis the largest prime.We consider Lagrange's theorem the so-called Mersenne number 2P-1 and show that any prime factor q If G is a finite(multiplicative)group of 2P-1 is bigger than p,which will yield the desired conclusion.Let g be and U is a subgroup.then U a prime dividing 2P-1,so we have 2P=1(mod g).Since p is prime,this divides G. means that the element 2 has order p in the multiplicative group Zo}of Proof.Consider the binary rela- the field Za.This group has q-1 elements.By Lagrange's theorem (see tion the box)we know that the order of every element divides the size of the a~b:←→ba-1∈U. group,that is,we have pq-1,and hence pk+1,and thus Steps above the functionf(t)= Pk ≤1+= 1 Pk-1 =1+ and therefore x( logx≤ k+1 =T(x)+1. k=1 Everybody knows that logz is not bounded,so we conclude that m()is unbounded as well,and so there are infinitely many primes

4 Six proofs of the infinity of primes Third Proof. Suppose P is finite and p is the largest prime. We consider the so-called Mersenne number 2p − 1 and show that any prime factor q of 2p − 1 is bigger than p, which will yield the desired conclusion. Let q be a prime dividing 2p − 1, so we have 2p ≡ 1 (mod q). Since p is prime, this Lagrange’s theorem If G is a finite (multiplicative) group and U is a subgroup, then |U| divides |G|. Proof. Consider the binary rela￾tion a ∼ b : ⇐⇒ ba−1 ∈ U. It follows from the group axioms that ∼ is an equivalence relation. The equivalence class containing an element a is precisely the coset Ua = {xa : x ∈ U}. Since clearly |Ua| = |U|, we find that G decomposes into equivalence classes, all of size |U|, and hence that |U| divides |G|. In the special case when U is a cyclic subgroup {a, a2,...,am} we find that m (the smallest positive inte￾ger such that am = 1, called the order of a) divides the size |G| of the group. In particular, we have a|G| = 1. means that the element 2 has order p in the multiplicative group Zq\{0} of the field Zq. This group has q − 1 elements. By Lagrange’s theorem (see the box) we know that the order of every element divides the size of the group, that is, we have p | q − 1, and hence p<q. Now let us look at a proof that uses elementary calculus. Fourth Proof. Let π(x) := #{p ≤ x : p ∈ P} be the number of primes that are less than or equal to the real number x. We number the primes P = {p1, p2, p3,...} in increasing order. Consider the natural logarithm log x, defined as log x =  x 1 1 t dt. 21 1 n n+1 Steps above the function f(t) = 1 t Now we compare the area below the graph of f(t) = 1 t with an upper step function. (See also the appendix on page 12 for this method.) Thus for n ≤ x<n + 1 we have log x ≤ 1 + 1 2 + 1 3 + ··· + 1 n − 1 + 1 n ≤  1 m, where the sum extends over all m ∈ N which have only prime divisors p ≤ x. Since every such  m can be written in a unique way as a product of the form p≤x pkp , we see that the last sum is equal to p∈P p≤x  k≥0 1 pk  . The inner sum is a geometric series with ratio 1 p , hence log x ≤ p∈P p≤x 1 1 − 1 p = p∈P p≤x p p − 1 = π (x) k=1 pk pk − 1 . Now clearly pk ≥ k + 1, and thus pk pk − 1 = 1+ 1 pk − 1 ≤ 1 + 1 k = k + 1 k , and therefore log x ≤ π (x) k=1 k + 1 k = π(x)+1. Everybody knows that log x is not bounded, so we conclude that π(x) is unbounded as well, and so there are infinitely many primes.

Six proofs of the infinity of primes J Fifth Proof.After analysis it's topology now!Consider the following curious topology on the set Z of integers.For a,bZ,b>0,we set Na.b={a+nb:n∈Z. Each set N is a two-way infinite arithmetic progression.Now call a set O CZ open if either O is empty,or if to every a E O there exists some b>0 with Na.b O.Clearly,the union of open sets is open again.If 01,02 are open,and a nO2 with Na.b O1 and Na.baO2. then a E Na.bbO1nO2.So we conclude that any finite intersection of open sets is again open.So,this family of open sets induces a bona fide topology on Z. Let us note two facts: (A)Any nonempty open set is infinite. (B)Any set Na.b is closed as well. Indeed,the first fact follows from the definition.For the second we observe b-1 Na.b =Z Nati.b; i=1 which proves that Na.b is the complement of an open set and hence closed. So far the primes have not yet entered the picture-but here they come. Since any number n1,-1 has a prime divisor p,and hence is contained in No.p,we conclude Z\{1,-1}No.p. "Pitching flat rocks,infinitely" pEp Now if P were finite,then No.p would be a finite union of closed sets (by (B)),and hence closed.Consequently,{1,-1 would be an open set, in violation of (A). ▣ Sixth Proof.Our final proof goes a considerable step further and demonstrates not only that there are infinitely many primes,but also that the seriesdiverges.The first proof of this important result was given by Euler(and is interesting in its own right),but our proof,devised by Erdos,is of compelling beauty. Let pi,p2,p3,...be the sequence of primes in increasing order,and assume that∑pee。 converges.Then there must be a natural numberk such that∑≥kt<子.Let us call p,,p the small primes,and Pk+1,Pk+2,...the big primes.For an arbitrary natural number N we there- fore find (1)

Six proofs of the infinity of primes 5 Fifth Proof. After analysis it’s topology now! Consider the following curious topology on the set Z of integers. For a, b ∈ Z, b > 0, we set Na,b = {a + nb : n ∈ Z}. Each set Na,b is a two-way infinite arithmetic progression. Now call a set O ⊆ Z open if either O is empty, or if to every a ∈ O there exists some b > 0 with Na,b ⊆ O. Clearly, the union of open sets is open again. If O1, O2 are open, and a ∈ O1 ∩ O2 with Na,b1 ⊆ O1 and Na,b2 ⊆ O2, then a ∈ Na,b1b2 ⊆ O1 ∩ O2. So we conclude that any finite intersection of open sets is again open. So, this family of open sets induces a bona fide topology on Z. Let us note two facts: (A) Any nonempty open set is infinite. (B) Any set Na,b is closed as well. Indeed, the first fact follows from the definition. For the second we observe Na,b = Z \ b −1 i=1 Na+i,b, which proves that Na,b is the complement of an open set and hence closed. “Pitching flat rocks, infinitely” So far the primes have not yet entered the picture — but here they come. Since any number n = 1, −1 has a prime divisor p, and hence is contained in N0,p, we conclude Z \ {1, −1} =  p∈P N0,p. Now if P were finite, then p∈P N0,p would be a finite union of closed sets (by (B)), and hence closed. Consequently, {1, −1} would be an open set, in violation of (A). Sixth Proof. Our final proof goes a considerable step further and demonstrates not only that there are infinitely many primes, but also that the series p∈P 1 p diverges. The first proof of this important result was given by Euler (and is interesting in its own right), but our proof, devised by Erdos, is of compelling beauty. ˝ Let p1, p2, p3,... be the sequence of primes in increasing order, and assume that p∈P 1 p converges. Then there must be a natural number k such that i≥k+1 1 pi < 1 2 . Let us call p1,...,pk the small primes, and pk+1, pk+2,... the big primes. For an arbitrary natural number N we there￾fore find  i≥k+1 N pi < N 2 . (1)

6 Six proofs of the infinity of primes Let N be the number of positive integers nN which are divisible by at least one big prime,and Ns the number of positive integers nN which have only small prime divisors.We are going to show that for a suitable N Nb+N。k+1 Let us now look at N.We write every nN which has only small prime divisors in the form n=ab,where an is the square-free part.Every an is thus a product of different small primes,and we conclude that there are precisely2 different square---free parts..Furthermore,.asbn≤√元≤√F, we find that there are at most vN different square parts,and so Ns≤2kVN Since (2)holds for any N,it remains to find a number N withN or1VN,and for this N =22k+2 will do. ☐ Appendix:Infinitely many more proofs Our collection of proofs for the infinitude of primes contains several other old and new treasures,but there is one of very recent vintage that is quite different and deserves special mention.Let us try to identify sequences S of integers such that the set of primes Ps that divide some member of S is infinite.Every such sequence would then provide its own proof for the infinity of primes.The Fermat numbers Fn studied in the second proof form such a sequence,while the powers of 2 don't.Many more examples are provided by a theorem of Issai Schur,who showed in 1912 that for every nonconstant polynomial p(x)with integer coefficients the set of all nonzero values{p(n)≠0：n∈N}is such a sequence..For the polynomial p(x)=z,Schur's result gives us Euclid's theorem.As another example, forp(x）=x2+1 we get that the“squares plus one”contain infinitely many different prime factors. The following result due to Christian Elsholtz is a real gem:It generalizes Schur's theorem,the proof is just clever counting,and it is in a certain sense Issai Schur best possible

6 Six proofs of the infinity of primes Let Nb be the number of positive integers n ≤ N which are divisible by at least one big prime, and Ns the number of positive integers n ≤ N which have only small prime divisors. We are going to show that for a suitable N Nb + Ns < N, which will be our desired contradiction, since by definition Nb + Ns would have to be equal to N. To estimate Nb note that N pi counts the positive integers n ≤ N which are multiples of pi. Hence by (1) we obtain Nb ≤  i≥k+1 N pi  < N 2 . (2) Let us now look at Ns. We write every n ≤ N which has only small prime divisors in the form n = anb2 n, where an is the square-free part. Every an is thus a product of different small primes, and we conclude that there are precisely 2k different square-free parts. Furthermore, as bn ≤ √n ≤ √ N, we find that there are at most √ N different square parts, and so Ns ≤ 2k √ N. Since (2) holds for any N, it remains to find a number N with 2k √ N ≤ N 2 or 2k+1 ≤ √ N, and for this N = 22k+2 will do. Appendix: Infinitely many more proofs Our collection of proofs for the infinitude of primes contains several other old and new treasures, but there is one of very recent vintage that is quite different and deserves special mention. Let us try to identify sequences S of integers such that the set of primes PS that divide some member of S is infinite. Every such sequence would then provide its own proof for the infinity of primes. The Fermat numbers Fn studied in the second proof form such a sequence, while the powers of 2 don’t. Many more examples Issai Schur are provided by a theorem of Issai Schur, who showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients the set of all nonzero values {p(n) =0: n ∈ N} is such a sequence. For the polynomial p(x) = x, Schur’s result gives us Euclid’s theorem. As another example, for p(x) = x2 + 1 we get that the “squares plus one” contain infinitely many different prime factors. The following result due to Christian Elsholtz is a real gem: It generalizes Schur’s theorem, the proof is just clever counting, and it is in a certain sense best possible.