Proofs from THE BOOK
Martin Aigner G¨unter M. Ziegler with illustrations by Karl H. Hofmann
Springer-Verlag Heidelberg/Berlin to appear August 1998 Preface
Paul Erd˝os 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. Erd˝os 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 Erd˝os’ 85th birthday. With Paul’s unfortunate death in the summer of 1997, he is not listed as a co-author. Instead this book is dedicated to his memory. Paul Erd˝os 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 Erd˝os 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 Erd˝os 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, Elke Pose, and J¨org 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 Erd˝os himself.
Berlin, March 1998 Martin Aigner Gunter¨ M. Ziegler Table of Contents
Number Theory 1
1. Six proofs of the infinity of primes :::::::::::::::::::::::::::::: 3
2. Bertrand’s postulate ::::::::::::::::::::::::::::::::::::::::::: 7
3. Binomial coefficients are (almost) never powers ::::::::::::::::: 13
4. Representing numbers as sums of two squares :::::::::::::::::::17
5. Every finite division ring is a field ::::::::::::::::::::::::::::: 23
6. Some irrational numbers ::::::::::::::::::::::::::::::::::::::27
Geometry 35
7. Hilbert’s third problem: decomposing polyhedra :::::::::::::::: 37
8. Lines in the plane and decompositions of graphs :::::::::::::::::45
9. The slope problem ::::::::::::::::::::::::::::::::::::::::::: 51
10. Three applications of Euler’s formula :::::::::::::::::::::::::: 57
11. Cauchy’s rigidity theorem :::::::::::::::::::::::::::::::::::::63
12. The problem of the thirteen spheres :::::::::::::::::::::::::::: 67
13. Touching simplices :::::::::::::::::::::::::::::::::::::::::::73
14. Every large point set has an obtuse angle ::::::::::::::::::::::: 77
15. Borsuk’s conjecture :::::::::::::::::::::::::::::::::::::::::: 83
Analysis 89
16. Sets, functions, and the continuum hypothesis :::::::::::::::::::91
17. In praise of inequalities ::::::::::::::::::::::::::::::::::::::101
18. A theorem of P´olya on polynomials :::::::::::::::::::::::::::109
19. On a lemma of Littlewood and Offord :::::::::::::::::::::::::117 VIII Table of Contents
Combinatorics 121
20. Pigeon-hole and double counting ::::::::::::::::::::::::::::: 123
21. Three famous theorems on finite sets ::::::::::::::::::::::::::135
22. Cayley’s formula for the number of trees :::::::::::::::::::::: 141
23. Completing Latin squares ::::::::::::::::::::::::::::::::::::147
23. The Dinitz problem ::::::::::::::::::::::::::::::::::::::::: 153
Graph Theory 159
25. Five-coloring plane graphs :::::::::::::::::::::::::::::::::::161
26. How to guard a museum :::::::::::::::::::::::::::::::::::::165
27. Tur´an’s graph theorem :::::::::::::::::::::::::::::::::::::::169
28. Communicating without errors ::::::::::::::::::::::::::::::: 173
29. Of friends and politicians :::::::::::::::::::::::::::::::::::: 183
30. Probability makes counting (sometimes) easy :::::::::::::::::: 187
About the Illustrations 196 Index 197 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. It shows that the sequence of primes
does not end.
fp ;::: ;p g r
Euclid’s Proof. For any finite set 1 of primes, consider
n = p p p +1 n p p
2 r
the number 1 .This has a prime divisor .But is
p p n
not one of the i : otherwise would be a divisor of and of the product
p p p n p p :::p = 1
2 r 1 2 r
1 , and thus also of the difference ,which
fp ;::: ;p g r is impossible. So a finite set 1 cannot be the collection of all
prime numbers.
= f1; 2; 3;:::g
Before we continue let us fix some notation. N is the set
= f::: ; 2; 1; 0; 1; 2;:::g
of natural numbers, Z the set of integers, and
= f2; 3; 5; 7;:::g P the set of primes. In the following, we will exhibit various other proofs (out of a much longer Lagrange’s Theorem list) which we hope the reader will like as much as we do. Although they
If G is a finite (multiplicative) group
use different view-points, the following basic idea is common to all of them:
jU j and U is a subgroup, then
The natural numbers grow beyond all bounds, and every natural number
Gj
divides j .
2 P n has a prime divisor. These two facts taken together force to be infinite. The next three proofs are folklore, the fifth proof was proposed by Proof. Consider the binary rela-
Harry F¨urstenberg, while the last proof is due to Paul Erd˝os. tion
1
a b : ba 2 U: The second and the third proof use special well-known number sequences.
It follows from the group axioms
P p
Second Proof. Suppose is finite and is the largest prime. We that is an equivalence relation.
p
1
consider the so-called Mersenne number 2 and show that any prime The equivalence class containing an
p
a
2 1 p
factor q of is bigger than , which will yield the desired conclusion. element is precisely the coset
p p
2 1 2 1 q p
Let q be a prime dividing ,sowehave mod .Since is
Ua = fxa : x 2 U g: p
prime, this means that the element 2 has order in the multiplicative group
Z nf0g Z q 1 q
q of the field . This group has elements. By Lagrange’s
Uaj = jU j Since clearly j ,wefind theorem (see the box) we know that the order of every element divides the
that G decomposes into equivalence
j q 1 p size of the group, that is, we have p , and hence . U j classes, all of size j , and hence U j jGj that j divides . n 2 F =2 +1 Third Proof. Next let us look at the Fermat numbers n for In the special case when U is a cyclic =0; 1; 2;::: n . We will show that any two Fermat numbers are relatively 2 m a; a ;::: ;a g prime; hence there must be infinitely many primes. To this end, we verify subgroup f we find the recursion that m (the smallest positive inte- m = 1 ger such that a , called the n 1 Y jGj order of a) divides the size of F = F 2 n 1 ; n k the group. k =0 4 Six proofs of the infinity of primes m = 3 F from which our assertion follows immediately. Indeed, if is a divisor of, 0 F F k < n m m = 1 2 F = 5 1 n say, k and ,then divides 2, and hence or .But F = 17 m =2 2 is impossible since all Fermat numbers are odd. F = 257 3 n n =1 F =3 To prove the recursion we use induction on .For we have 0 F = 65537 4 F 2= 3 and 1 . With induction we now conclude F = 641 6700417 5 n n 1 Y The first few Fermat numbers Y F = F F = F 2 F = k k n n n k =0 k =0 n n n+1 2 2 2 = 2 1 2 +1 = 2 1 = F 2: n+1 Now let us look at a proof that uses elementary calculus. x :=fp x : p 2 Pg Fourth Proof. Let be the number of primes that are less than or equal to the real number x. We number the primes P = fp ;p ;p ;:::g 2 3 1 in increasing order. Consider the natural logarithm R x 1 x log x = dt log , defined as . 1 t 1 1 t = Nowwecomparetheareabelowthegraphoff with an upper step t function. (See also the appendix on page 10 for this method.) Thus for x n we have 1 1 1 1 log x 1+ + + ::: + + 2 3 n 1 n 1 2 n X 1 1 t = Steps above the function f ; m 2 N t where the sum extends over all which have m x only prime divisors p . Since every such m can be written in a unique way as a product of the form Q k p p , we see that the last sum is equal to px Y X 1 : k p p2P k 0 px The inner sum is a geometric series with ratio 1 , hence p x Y Y Y p 1 p k = log x = : 1 p 1 p 1 1 k p p2P p2P k =1 px px p k +1 Now clearly k , and thus p 1 1 k +1 k = 1+ 1+ = ; p 1 p 1 k k k k and therefore x Y k +1 log x = x +1: k k =1 x x Everybody knows that log is not bounded, so we conclude that is unbounded as well, and so there are infinitely many primes. Six proofs of the infinity of primes 5 Fifth Proof. After analysis it’s topology now! Consider the following a; b 2 Z b>0 curious topology on the set Z of integers. For , we set N = fa + nb : n 2 Zg: a;b N Each set a;b is a two-way infinite arithmetic progression. Now call a set Z O a 2 O O open if either is empty, or if to every there exists some b > 0 N O with a;b . Clearly, the union of open sets is open again. If O ;O a 2 O \ O N O N O 2 1 2 a;b 1 a;b 2 1 are open, and with and , 1 2 a 2 N O \ O b 1 2 then a;b . So we conclude that any finite intersection 1 2 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 non-empty open set is infinite. N (B) Any set a;b is closed as well. Indeed, the first fact follows from the definition. For the second we observe b 1 [ N = Zn N ; a;b a+i;b i=1 N which proves that a;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. 6=1; 1 p Since any number n has a prime divisor , and hence is contained N ;p in 0 , we conclude [ nf1; 1g = N : Z “Pitching flat rocks, infinitely” 0;p p2P S P N ;p Now if were finite, then 0 would be a finite union of closed sets p2P 1; 1g (by (B)), and hence closed. Consequently, f 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 P the series 1 diverges. The first proof of this important result was p2P p given by Euler (and is interesting in its own right), but our proof, devised by Erd˝os, is of compelling beauty. p ;p ;p ;::: 2 3 Let 1 be the sequence of primes in increasing order, and P 1 assume that converges. Then there must be a natural number k p2P p P 1 1 p ;::: ;p < k such that . Let us call 1 the small primes, and ik +1 p 2 i p ;p ;::: N +1 k +2 k the big primes. For an arbitrary natural number we therefore find X N N < : (1) p 2 i ik +1 6 Six proofs of the infinity of primes N n N Let b be the number of positive integers which are divisible by at N n N least one big prime, and s the number of positive integers which have only small prime divisors. We are going to show that for a suitable N N + N < N; b s N + N s which will be our desired contradiction, since by definition b would have to be equal to N . N N b c n N To estimate b note that counts the positive integers which p i p are multiples of i . Hence by (1) we obtain k j X N N < : N b (2) p 2 i ik +1 N n N Let us now look at s . We write every which has only small prime 2 n = a b a a n divisors in the form ,where n is the square-free part. Every n n is thus a product of different small primes, and we conclude that there are p p k 2 b n N precisely different square-free parts. Furthermore, as n , p we find that there are at most N different square parts, and so p k N 2 N: s p N k N 2 N Since (2) holds for any N , it remains to find a number with 2 p k +1 2k +2 N N =2 or 2 , and for this will do. References P [1] P. ERDOS˝ : Uber¨ die Reihe 1 , Mathematica, Zutphen B 7 (1938), 1-2. p [2] L. EULER: Introductio in Analysin Infinitorum, Tomus Primus, Lausanne 1748; Opera Omnia, Ser. 1, Vol. 90. [3] H. FURSTENBERG¨ : On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353.