Proofs from the Book

219 Pages·2009·7.924 MB·English

by  Martin Aigner, Günter M. Ziegler, Karl H. Hofmann

Martin Aigner
Giinter M. Ziegler
Proofs from
THE BOOK
Third Edition
With 250 Figures
Including Illustrations
by Karl H. Hofmann
Springer
Preface
PaulErdo˝slikedtotalkaboutTheBook,inwhichGodmaintainstheperfect
proofsformathematicaltheorems,followingthedictumofG.H.Hardythat
thereisnopermanentplaceforuglymathematics.Erdo˝salsosaidthatyou
need not believe in God but, as a mathematician, you should believe in
TheBook. A fewyearsago, we suggestedto him to writeup afirst (and
very modest) approximationto The Book. He was enthusiastic aboutthe
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 Erdo˝s’ 85th birthday. With Paul’s unfortunatedeath
inthesummerof1997,heisnotlistedasaco-author. Insteadthisbookis
dedicatedtohismemory.
PaulErdo˝s
Wehavenodefinitionorcharacterizationofwhatconstitutesaprooffrom
The Book: all we offer here is the examples that we have selected, hop-
ingthatourreaderswillshareourenthusiasmaboutbrilliantideas, clever
insights and wonderful observations. We also hope that our readers will
enjoythisdespitetheimperfectionsofourexposition.Theselectionistoa
greatextentinfluencedbyPaulErdo˝shimself.Alargenumberofthetopics
weresuggestedbyhim,andmanyoftheproofstracedirectlybacktohim,
or wereinitiated byhis supremeinsightin askingtherightquestionor in
makingtherightconjecture.Sotoalargeextentthisbookreflectstheviews
ofPaulErdo˝sastowhatshouldbeconsideredaprooffromTheBook. “TheBook”
Alimitingfactorforourselectionoftopicswasthateverythinginthisbook
is supposed to be accessible to readers whose backgrounds include only
a modest amount of technique from undergraduatemathematics. A little
linearalgebra,somebasicanalysisandnumbertheory,andahealthydollop
of elementary conceptsand reasoningsfrom discrete mathematicsshould
besufficienttounderstandandenjoyeverythinginthisbook.
We are extremely gratefulto the many people who helped and supported
us with this project — among them the students of a seminar where we
discussedapreliminaryversion,toBennoArtmann,StephanBrandt,Stefan
Felsner, Eli Goodman, Torsten Heldmann, and Hans Mielke. We thank
MargritBarrett,ChristianBressler,EwgenijGawrilow,ElkePose,andJo¨rg
Rambau for their technicalhelp in composingthis book. We are in great
debt to Tom Trotter who read the manuscript from first to last page, to
Karl H. Hofmann for his wonderfuldrawings, and most of all to the late
greatPaulErdo˝shimself.
Berlin,March1998 MartinAigner Gu¨nterM.Ziegler
(cid:1)
Preface to the Second Edition
The first edition of this book got a wonderful reception. Moreover, we re-
ceived an unusual number of letters containing comments and corrections,
some shortcuts, as well as interesting suggestions for alternative proois and
new topics to treat. (While we are trying to record pelfect proofs, our
exposition isn't.)
The second edition gives us the opportunity to present this new version of
our book: It contains three additional chapters, substantial revisions and
new proofs in several others, as well as minor amendments and improve-
ments, many of them based on the suggestions we received. It also misses
one of the old chapters, about the "problem of the thirteen spheres," whose
proof turned out to need details that we couldn't complete in a way that
would make it brief and elegant.
Thanks to all the readers who wrote and thus helped us - among them
Stephan Brandt, Christian Elsholtz, Jurgen Elstrodt, Daniel Grieser, Roger
Heath-Brown, Lee L. Keener, Christian Lebceuf, Hanfried Lenz, Nicolas
Puech, John Scholes, Bernulf WeiBbach, and many others. Thanks again
for help and support to Ruth Allewelt and Karl-Friedrich Koch at Springer
Heidelberg, to Christoph Eyrich and Torsten Heldmann in Berlin, and to
Karl H. Hofmann for some superb new drawings.
Berlin, September 2000 Martin Aigner . Giinter M. Ziegler
Preface to the Third Edition
We would never have dreamt, when preparing the first edition of this book
in 1998, of the great success this project would have, with translations into
many languages, enthusiastic responses from so many readers, and so many
wonderful suggestions for improvements, additions, and new topics - that
could keep us busy for years.
So, this third edition offers two new chapters (on Euler's partition identities,
and on card shuffling), three proofs of Euler's series appear in a separate
chapter, and there is a number of other improvements, such as the Calkin-
Wilf-Newman treatment of "enumerating the rationals." That's it, for now!
We thank everyone who has supported this project during the last five
years, and whose input has made a difference for this new edition. This
includes David Bevan, Anders Bjorner, Dietrich Braess, John Cosgrave,
Hubert Kalf, Gunter Pickert, Alistair Sinclair, and Herb Wilf.
Berlin, July 2003 Martin Aigner . Giinter 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 .................1 3
4 . Representing numbers as sums of two squares ...................1 7
5 . Every finite division ring is a field .............................2. 3
6. Some irrational numbers .....................................2.7
7 . Three times 7r2/6 ............................................3.5
Geometry 43
8. Hilbert's third problem: decomposing polyhedra .................4 5
9 . Lines in the plane and decompositions of graphs .................5 3
10. The slope problem ............................................ 59
11 . Three applications of Euler's formula ..........................6 5
12. Cauchy's rigidity theorem ....................................7.1
13. Touching simplices ..........................................7.5
14. Every large point set has an obtuse angle .......................7 9
15. Borsuk's conjecture ..........................................8 5
Analysis 9 1
16. Sets. functions. and the continuum hypothesis ...................9 3
17 . In praise of inequalities .....................................1.0 9
18. A theorem of Pdya on polynomials ...........................1 17
19. On a lemma of Littlewood and Offord ........................1. 23
20 . Cotangent and the Herglotz trick .............................1.2 7
21 . Buffon's needle problem .....................................1 33
VIII Table of Contents
Corn binatorics 137
22 . Pigeon-hole and double counting .............................1 39
23 . Three famous theorems on finite sets ..........................1 51
24 . Shuffling cards .............................................1.5 7
25 . Lattice paths and determinants ...............................1.6 7
26 . Cayley's formula for the number of trees ......................1 73
27 . Completing Latin squares ....................................1 79
28 . The Dinitz problem .........................................1.8 5
29 . Identities versus bijections ...................................1 91
Graph Theory 197
30. Five-coloring plane graphs ..................................1. 99
3 1. How to guard a museum ....................................2.0 3
32 . Turhn's graph theorem ......................................2.0 7
33. Communicating without errors ...............................2 13
34 . Of friends and politicians ...................................2.2 3
35. Probability makes counting (sometimes) easy ..................2 27
About the Illustrations 236
Index 237
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
doesnotend.
Euclid’s Proof. For any finite set of primes, consider
t(cid:4)he number . This ph1a;s::a:p;prirme divisor . But is
f g
notoneofthne=:po1tph2erwipser +w1ouldbenadivisorof andof tpheprodpuct
(cid:1)(cid:1)(cid:1)
, andpithus also ofpthe difference n , which
pis1pim2pospsirble. So a finite set cannnotpb1pe2th::e:cporlle=cti1on ofall
(cid:1)(cid:1)(cid:1) (cid:0)
primenumbers. p1;::: ;pr
f g (cid:3)
Beforewe continuelet usfix somenotation. is theset
ofnaturalnumbers, N = th1e;2se;t3o;:f:i:ntegers,and
Zth=ese:t:o:f;pri2m;es.1;0;1;2;::: f g
P= 2;3;5;7;::: f (cid:0) (cid:0) g
Inthefollowing,wewillexhibitvariousotherproofs(outofamuchlonger
f g Lagrange’sTheorem
list)whichwehopethereaderwilllikeasmuchaswedo. Althoughthey
If isafinite(multiplicative)group
usedifferentview-points,thefollowingbasicideaiscommontoallofthem:
anGd is a subgroup, then
The natural numbers grow beyond all bounds, and every natural number
divideUs . jUj
has a prime divisor. These two facts taken together force to be jGj
ninfinit2e. Thenextthreeproofsarefolklore,thefifthproofwaspropoPsedby Proof. Consider the binary rela-
(cid:21) (cid:4)
tion
HarryFu¨rstenberg,whilethelastproofisduetoPaulErdo˝s.
(cid:0)1
Thesecondandthethirdproofusespecialwell-knownnumbersequences. a(cid:24)b:() ba 2U:
It follows from the group axioms
Second Proof. Suppose is finite and is the largest prime. We that is an equivalence relation.
(cid:4)considertheso-calledMersennPe number p and showthatanyprime Thee(cid:24)quivalenceclasscontainingan
p
factor of isbiggerthan ,whichw2illyi1eldthedesiredconclusion. element ispreciselythecoset
Let bqea2pprime1dividing p, so wehav(cid:0)e mod . Since is a
(cid:0) p p
primqe,thismeansthatthee2lemen1t hasorder 2inthe1m(ultipqli)cativegropup
(cid:0) (cid:17) Ua=fxa:x2Ug:
of the field . This grou2p has p elements. By Lagrange’s
Since clearly , we find
tZhqenofre0mg (seetheboxZ)wq eknowthattheorqde(cid:0)ro1feveryelementdividesthe that decompjUosaejsi=ntojUeqjuivalence
sizeofthegroup,thatis,wehave ,andhence . classGes, all of size , and hence
pjq(cid:0)1 p<q (cid:3) that divides .jUj
ThirdProof. NextletuslookattheFermatnumbers n for jUj jGj (cid:3)
(cid:4) . WewillshowthatanytwoFermatnumbFenrs=ar2e2rel+ati1vely Inthespecialcasewhen isacyclic
pnri=me0;;h1e;n2c;e::t:heremustbeinfinitelymanyprimes. Tothisend,weverify subgroup 2 mU we find
therecursion that (thfea;sama;l:le:s:t;aposgitive inte-
m
ger such that , called the
m
order of ) diavides=th1e size of
n(cid:0)1
thegroupa. jGj
Fk = Fn 2 (n 1);
Y (cid:0) (cid:21)
k=0
4 Sixproofsoftheinfinityofprimes
fromwhichourassertionfollowsimmediately.Indeed,if isadivisorof,
F0 = 3 say, and , then divides 2, and hence m or . But
F1 = 5 FkisimpFonss(ikble<sinnc)eallFemrmatnumbersareodd. m = 1 2
F2 = 17 m=2
Toprovetherecursionweuseinductionon . For wehave
F3 = 257
and . Withinductionwenowconnclude n = 1 F0 = 3
F4 = 65537
F1 2=3
F5 = 641(cid:1)6700417 (cid:0)
ThefirstfewFermatnumbers
n n(cid:0)1
Fk = Fk Fn = (Fn 2)Fn =
Y (cid:16)Y (cid:17) (cid:0)
k=0 k=0
n n n+1
=(22 1)(22 +1) = 22 1 = Fn+1 2: (cid:3)
(cid:0) (cid:0) (cid:0)
Nowletuslookataproofthatuseselementarycalculus.
FourthProof. Let bethenumberofprimes
t(cid:4)hat are less than or eq(cid:25)u(axl)t:o=th#e rpeal nxum:pberP. We number the primes
f (cid:20) 2 g
in increasing order. Conxsider the natural logarithm
P =,dpe1fi;npe2d;pa3s;::: .
f g x 1
1 lNoogwxwecomparelothgexa=reaR1beltodwt thegraphof withanupperstep
1
function. (See also the appendix on page 10ff(otr)t=histmethod.) Thus for
wehave
n x<n+1
(cid:20)
1 1 1 1
logx 1+ + +:::+ +
(cid:20) 2 3 n 1 n
Steps1abo2vethefunction 1n 1 wherethesum(cid:0)extendsoverall whichhave
f(t)= t ; onlyprimedivisors . m N
(cid:20) Xm 2
p x
Sinceeverysuch canbewritteninaunique(cid:20)wayasaproductoftheform
,weseethmatthelastsumisequalto
kp
p
pQ(cid:20)x
1
:
Y (cid:16)X pk(cid:17)
p2P k(cid:21)0
p(cid:20)x
Theinnersumisageometricserieswithratio ,hence
1
p
(cid:25)(x)
1 p pk
logx (cid:20) pY2P 1(cid:0) p1 = pY2P p(cid:0)1 = kY=1 pk(cid:0)1:
p(cid:20)x p(cid:20)x
Nowclearly ,andthus
pk k+1
(cid:21)
pk 1 1 k+1
= 1+ 1+ = ;
pk 1 pk 1 (cid:20) k k
andtherefore (cid:0) (cid:0)
(cid:25)(x)
k+1
logx = (cid:25)(x)+1:
(cid:20) Y k
k=1
Everybodyknows that is not bounded, so we concludethat is
unboundedaswell,andlsoogtxhereareinfinitelymanyprimes. (cid:25)(x)
(cid:3)
Sixproofsoftheinfinityofprimes 5
FifthProof. Afteranalysisit’stopologynow! Considerthefollowing
(cid:4)
curioustopologyontheset ofintegers.For , weset
Z a;b Z b>0
2
Na;b = a+nb:n Z :
f 2 g
Eachset isatwo-wayinfinitearithmeticprogression. Nowcallaset
oNpean;bif either is empty, or if to every thereexists some
O Zwith O. Clearly, the union of opaen seOts is open again. If
(cid:18) 2
b > 0 are oNpean;b, andO with and ,
(cid:18)
Oth1e;nO2 a O1. SoOw2econcNluad;eb1thataOn1yfiniteNian;bte2rsectOio2n
2 \ (cid:18) (cid:18)
ofopaenseNtsai;sb1abg2ainOop1en.OSo2,thisfamilyofopensetsinducesabonafide
2 (cid:18) \
topologyon .
Z
Letusnotetwofacts:
(A) Anynon-emptyopensetisinfinite.
(B) Anyset isclosedaswell.
Na;b
Indeed,thefirstfactfollowsfromthedefinition.Forthesecondweobserve
b(cid:0)1
Na;b = Z Na+i;b;
n[
i=1
whichprovesthat isthecomplementofanopensetandhenceclosed.
Na;b
So far the primeshavenotyetenteredthe picture— but heretheycome.
Sinceanynumber hasaprimedivisor ,andhenceiscontained
in ,weconclunde=1; 1 p
6 (cid:0)
N0;p
“Pitchingflatrocks,infinitely”
Z 1; 1 = N0;p:
nf (cid:0) g p[2P
Nowif werefinite,then wouldbeafiniteunionofclosedsets
(by(B))P,andhenceclosed. pC2oPnNse0q;puently, wouldbeanopenset,
S
inviolationof(A). 1; 1
f (cid:0) g (cid:3)
Sixth Proof. Our final proof goes a considerable step further and
(cid:4)
demonstratesnot only that there are infinitely many primes, but also that
the series diverges. The first proof of this important result was
1
givenbyEulepr2P(apndisinterestinginits ownright), butourproof,devised
P
byErdo˝s,isofcompellingbeauty.
Let be the sequence of primes in increasing order, and
assump1e;pth2a;tp3;::: converges. Then there must be a natural number
1
such that p2Pp . Let us call the small primes, ankd
P 1 1
i(cid:21)k+th1epibi<g p2rimes. For anp1a;rb:i:t:ra;rpyknatural number we
P
pthke+r1e;foprke+fi2n;:d:: N
(1)
N N
< :
X pi 2
i(cid:21)k+1
6 Sixproofsoftheinfinityofprimes
Let bethenumberofpositiveintegers whicharedivisiblebyat
leastNobnebigprime,and thenumberofnposiNtiveintegers which
(cid:20)
haveonlysmallprimedivNissors.WearegoingtoshowthatfonrasuNitable
(cid:20)
N
Nb+Ns < N;
whichwillbeourdesiredcontradiction,sincebydefinition would
havetobeequalto . Nb+Ns
N
To estimate notethat countsthe positiveintegers which
N
aremultiplesNobf . Hencebpyi (1)weobtain n N
b c (cid:20)
pi
(2)
N N
Nb < :
(cid:20) X jpik 2
i(cid:21)k+1
Letusnowlookat . Wewriteevery whichhasonlysmallprime
divisorsintheformNs ,where n isNthesquare-freepart. Every
2 (cid:20)
isthusaproductofdnif=feraenntbnsmallprimanes,andweconcludethatthereaarne
precisely differentsquare-freeparts. Furthermore,as ,
k
wefindth2atthereareatmost differentsquareparts,banndsopn pN
(cid:20) (cid:20)
pN
k
Ns 2 pN:
(cid:20)
Since(2)holdsforany ,itremainstofindanumber with
k N
or ,andforNthis willdo. N 2 pN 2
2k+1 pN N =22k+2 (cid:20) (cid:3)
(cid:20)
References
[1] P.ERDO˝S:U¨berdieReihe 1,Mathematica,ZutphenB7(1938),1-2.
Pp
[2] L. EULER: Introductio in Analysin Infinitorum, Tomus Primus, Lausanne
1748;OperaOmnia,Ser.1,Vol.90.
[3] H. FU¨RSTENBERG: On the infinitude of primes, Amer. Math. Monthly 62
(1955),353.
Bertrand's postulate Chapter 2
We have seen that the sequence of prime numbers 2,3,5,7,.. . is infinite.
To see that the size of its gaps is not bounded, let N := 2 . 3 . 5 . . . . . p
+
denote the product of all prime numbers that are smaller than k 2, and
note that none of the k numbers
< +
is prime, since for 2 5 i k 1 we know that i has a prime factor that is
+ +
smaller than k 2, and this factor also divides N, and hence also N i.
With this recipe, we find, for example, for k = 10 that none of the ten
numbers
2312,2313,2314.. . . ,2321
is prime.
But there are also upper bounds for the gaps in the sequence of prime num-
bers. A famous bound states that "the gap to the next prime cannot be larger
than the number we start our search at." This is known as Bertrand's pos-
tulate, since it was conjectured and verified empirically for n < 3 000 000
by Joseph Bertrand. It was first proved for all n by Pafnuty Chebyshev in
Joseph Bertrand
1850. A much simplcr proof was given by the Indian genius Ramanujan.
Our Book Proof is by Paul Erdiis: it is taken from Erdiis' first published
paper, which appeared in 1932, when Erd6s was 19.
4
Bertrand's postulate.
Beweis eines Satzes von Tschebyschef.
>
For every n 1, there is some prime number p with n < p 5 272. Vun P Enods ~n Budapest.
(2)
Proof. We will estimate the size of the binomial coefficient care-
fully enough to see that if it didn't have any prime factors in the range
<
n < p 271, then it would be "too small." Our argument is in five steps.
(1) We first prove Bertrand's postulate for n < 4000. For this one does not
need to check 4000 cases: it suffices (this is "Landau's trick") to check that
is a sequence of prime numbers, where each is smaller than twice the previ-
ous one. Hence every interval {y : n < y 5 2n), with n 5 4000, contains
one of these 14 primcs.