OTHER
CIlAYWCK
PUBLICATIONS
PONTRYAGIN:
Foundation., of COllbinator i a I To polo g y
NO\OZHILOV:
Founda t ions of the \,." I ineor Th'ory of Elastici Iy
ALEKSA'lIOROV:
Combtnatorial Va I. 1
Topology,
TlIH
E:E
PEAHLS
N U \1 U E H
o F
T /I E 0 H Y
BY
A. Y. KIIINCIIIN
G RAY L 0 C K
PRE S S
ROCHESTEH., N. Y.
1952
OTHER
GR4YWCK
PUBLI CA nONS
PONTRYAGIN:
Foundattons of Co.~ina tor i a I To polo g y
NOV07.IIILu\':
F 0 u n d a t ion s 0 f t I,· v"" I t n e ar Theory of Elasticity
ALEKSANDROV:
Combtnatorial Va 1. 1
Topology,
T H R E E
N U
~I
PEARLS
TilE 0
BElt
o
n y
BY
A. ). KIIINCHIN
GRAYLOCK
PRE S S
ROCHESTEH, N. Y. 1952
F
TRANSLATED FROM
Tin:
SECOND (1948), REVISED RUSSIAN EDITION
BY F. [HGEMIRL
H.
KLlM~1
W. SEIDEL
Copyright, 1952, by
GHA Y LOCK PHESS Rochester, N. Y.
Second Print in g, 1956 All rights reserved. This book. or parts thereof, may nol be reproduced in any form, or IrBnslet .. d, without permission in "riting from th e pu blish ers.
Manufactured in the United States of America
FOREWOHD
Thi~ little hook is devoted to three lheorems in arithmetic, y,laich, in spite of their apparent simplicity, have been the ob.iects of the efforts of many important mathematical scholars. The proofs \~hicl, ilre presented here make use of completely elementary means, (although lhey are not very simple).
The book can be unden,lood by beginning college students, and is intended for wide circle.,; of lovers of mathematics.
CON T
A
LETTJo:II
CIIAPT[':R
TO I.
TilE: VAN
FnO~T
!)ER
CHAPTER Ill.
NT S
(IN
LIEU
I\.\ERDE:-f'S
A"ITHMt~TIC
CII.\PT.:R II.
j<~
PRI)(~n
01·
LAN!) 'l.IJ-SC!lf';IHF.L'1ANN MANN'S
AI';
IIYPOTIIESIS
18
TIIEOR.E'1
ELt:'1[~N'IARY
SOI.UTION
U1\
11
r."SIO'<::;
AND
9
Pln:FO\CE)
Tln~OREM
THE
PROBLF.M
A
OF
WAlliNG'S
37
A LETTER TO TilE FRONT (IN
LIEU
OF
A
PREFACE)
March 24, 1945 Dear Seryozlta, lour lettE'r sent from the hospital gave me threefold pleasure. First of all, your reque!'>t that J sell'! YOll "o;;ome little mathematical pearls" showed that you arc really getting well, and arc not merely trying, as a brave fighter, to relieve your friends' anxiety. That \Va."! my first pleasure. Furthermore, you gdve me occasion to reflect on ho", it is that in this war such young fighter!'! as you happen to pursue their fti\orite occupntion-the occupation which they chprished dlreudy before the war, and frOIl1 which the "'ar ha~ tOni them-so PdSsionately during every little respttp. There \\ill:> nothing like this during the last \\orlol Wilf. In those days a young man who had urrived at tltp ffront almost invariablv felt that his life had been disrurted, that what had been the s'uhslance of his life before had become for him an unrealizable legend. ;-"ow, however, there are some who "'Tite dissertations in the intervals between battles, antI defend them on their return during a brief furlough! Is it not because you feel with your ",hole being, that your accomplishments in war and in your favorite occupations-science. art, practical activity-are two links of one and the same great cuu!'le? And if so, is not this feeling, perhaps, one of the mainsprings of your victories which we, here at home, are so enthusiastic about? This thought gratified me very much. and that was my second pleasure. And so I began to think about what to send you. I do not know you very well-you attended my lectures for only one year. Nevertheless I retained a finn conviction of your profound and serious attitude toward science, and therefore I did not want to send you merely some trinkets which were showy hut of little substance scientifically. On the other hand I knew that your preparation was not very great-you spent only one year in the university classroom, dnd during three years of uninterrupted service at the Front YOIl will hardly have had time to study. After several days' delibera-
10 tion I have made a choice. You must judge for yourself whether it is a happy one or not. Personally I considl'r the three theorems of arithmetic which I am sending you, to be ~cnuinp PC..lris of our science. From time t.o time, remarkablt,. curious pl'oLll'lIIs turn up in arithmetic, this ol(lest, bUL forever youthful. branch of mathematic!-<. In content they are so elementary that dIly schuolboy can understand them. They are usually eoncernerl with the proof of some very simple law governing the world of numhC'Ts, a JdW which turns out to be correct in all tested special cases. The problem now is Lo prove thaL it is in fact always correct. And yet, in spite of the apparent simplicity of the problem. ils solution resists, for years and sometimes centuries, the efforts of the most important scholars of the age. \ou mllst drlmit that this is extraordinarily tempting. I have selet:ted thrp!" slich I'Tf)blC'ms for you. They have all been solved quite recently, ,lnd thprr- aI'e 1\\0 remarkable common features in their history. First, all three problcms have been solved by the most I"lementary arithllH'ti (ill IlIcthnds (do not, howC'ver, confuse elementary wilh simple: as }llU will Sf''', the ""lulions of nil three problems are not very simple, and it will require nnt a little effort on your parI Lo IIndprstand thf'lII ",,11 and a,.."imiiate the-m). Secondly, all three problems have Lp-en slllvpd Ly vcr)' yllllng, Le-' ginning mathematicians, youths of hardly your .J.~e. aftf'r it series of unsuccessful attacks on the part of "venerable" scholars. Isn't this a spur full of promise for future scholars like YllU? \\hal an encouraging call to sCientifiC' daring! The work of expounding these theorellls cllmpelleoi lilt> to penetrate morl! deeply inlo the structure (If their lTIugnifiel'llt proofs, dnd gave me greaL pleasure. That was my third pleasurp-,
r wish you
the best of success-in eomuut ann in sCience. ) ours,
4, f...hinchin
ell APTER I
\ A:--i DF.R WAERDE:!'l'S TIIJ..:OHE\\ ON ARITII\1F.T[C PHOGH ~SSIONS ~1
In the summer of 1921-1 I spent several weeks in Goltingen. ,h usual, many forei/>,'1l scholars hud arrived there fOf the summer semester. I got to know nUiny of them, and actually nu\(le friends with some. /\L the time of my urrival. tht' topic of the day was the bri!lidnt fesult of a youn~ Hollander, vun rlcr W
12 Quite recently, M. A. Lukomskaya Cof \/finsk} discovered and sent me a considerably simpler and more transparent proof of van der Waerden's theorem, which, with her kind permission, J am going to show you in what follow~.
§2 Actually, van der Waerden proved somewhat more than what was required. In the first place, he assumes that the natural numbers are divided, not into two, but into arbitrarily many, say k, clas~es (sets). In the second place, it turns out that it is not necessary to decompose the entire sequence of natural numbers in order to guarantee the existence of an arithmetic progression of prescribed {arbitrarily large} length I in at least one of these classes; a certRin segment of it suffices for this purpose. The length, n (k, I), of this segment is a function of the numbers k and l. Of course it doesn't matter where we take this segment, so long as there are n (k, l) successive natural numbers. Accordingly, vun cler \\uerden's theorem can be formulated as follows: Let k and l be tu,o arbitrary natural number". Then there exists a natural number n(k, l) such that, if an arbitrary segment, 'of length n(k, l), of the sequence of natural numbers is divided in any maT/ner into k classes (some of u;hich may be empty), t!ten em arithmetic progression of length l appears tn at least one of t!tese classes. This theorem is true trivially for 1='2. To scc this, it suffices to set n (k, 2) = k + 1: for if k + 1 numLcr's arc JiviJcd int" k classes, then certainly at least one of these classes contains more than one number, and an arLitrary pair of numbers forms an arithmetit progression of lcngth 2, which proves the theorem. \\P shull prov(' the theorem by induction on l. Consequently, ¥.e shall assume throughout the following that the theorem has already been verified for some number l ~ 2 and for arhitrary val ueb of k, and shall sho¥. thaI it retains its v;lidity for tbe number 1+1 (anrl natllrdlly als.) for all values of k).
§3 According to our assumption, then, for every natural nllmber k there is a natural number n (k, l) such that, if an arbitrary segment, of length n(k, l), of the natural numbers, is divided in any manner into k classes, there exis~s in at least one of these t:lIlSSI'S an arithmetic progression of length l. ~e must thcn prow that. for every natural number k, an n(k, l+ 1) also exists. \\p solve this problem by actually constructing the number ",(I.:.l+ 1). ,[,,, this end we set
13 qo",.l, no=n(k,l)
and then define the numbers q1' q'2' ... , n1' n:;>, .•. successivcly as follows: If qs-l and n s _l have already been defined for some 5 >0, we put n s =n(k\l)
(5=1,2, ... ).
The numbers n s ' qs are obviously defined hereby for an arbitrary s ~O. \\c now assert thdt for n(k, l+ 1) we may take the number qk. \\ e have to show then that if 11 segment, of length qk' of the sequence of natural numbers is divided in any manner into k classes, then there is an arithmetic progression of length l+ 1 in at least one of these classes. The remainder of the chaptPf is devoted to this proof. In the sequel we set l+l=l'for brevity.
§4 Suppose then thdt the segment _\. of length qk' of the sequence of natural numbers is divided in an arbitrary "'ay into k classes. We say that two numbers a and b of \ are of the same type, if a and b belong to the same class, and we then write a".b. Two equally long subsegments of t\, 8=(a,a+ l, ... ,a+r) and B'=(a~a'+l, ... ,a'+r), are said to be of the Sdlllf" type, if a".a~ a+1".a'+ 1, .... a+r".a'+r, and we then write 8".8'. The number of different possible Iypes for the numbers of the segment .\ is obviously equal to k. For seg",!ents of the form (a, a+ 1) (j. e., for segments of length 2) the number of possible types is k'2; and in general, for segments of length m, it is km. (Of course not all these types need actually appear in the segment ~.) Since qk=2nk-lqk_l (see (l», the segment l\ can be regarded as a sequence of 2n k _l subsegments of length qk.l' Such subsegments, as we have just seen, can have k qk - l different types. The left half of the segment t\ now contains nk_l such subsegments, where nk_l=n(kqk.l,l) according to (1). Because of the meaning of the number n(k q k-l, l), we can assert* that the left half of the segment \ contains an arithmetic progression of l of these subsegments of ·Work with the initial numbers of the
nk_l
subsegments. ffrans.)
14 the same type,
'\1' \2, .... \/ of length qk-l; here we day for brevity that equally long segnu'nts \ form an arithmetic progression, if their initial numbE'rs form such a progression. We call the difference between the initial numhers of two neighboring segments of the progression \ 10 \ 2, •••• ~\1 the difference d 1 of this progression. Naturally the difference between the second (or third, fourth, etc.) numbers of two such neighboring segments is likewise equal to d l ' To this progression of segments we now add the succeeding, (i+1)-st, term \l'(we recall that l'=l+1) which may already project beyond the boundary of the left half of the segment ,1, hut which in any case still belongs entirely to the segment \. The segments "1>;\2' ... , \/,!\/' then form an arithmetic progJ.'ession, of length l'=l+] and difference dlo of sewnents of length qk-l' where A 1 ':\2, ... ,I\l are of the same type. \\'e know nothing about the type of the last segment A l " This completes the first step of our construction. [t would be well if you thought it through once more before we continued. ~5
We now proceed to the second step. We tuke an arbitrary one of the first l terms of the progression of segments just constructed. Let this term be \1' so that l::;;i1::;;l; "i1 is a segment of length qk-l' We treat it the same way as we treated the segment _1. Since qk-l =2n k _2Qk_2' the left half of the segment :~i1 can be regarded as a sequence of n k _2 subsegments of length Qk-2' Forsubsegments of this length there are k qk-2 types possible, and on the other hand nk_2=n(k Qk - 2 ,l) because of (]). Therefore the left half of -\1 must contain a progression of l of these subsegments of the same type, Ai1i2 (I::;;i2~l), of length qk.2' Let d 2 be the difference of this progression 0. c., the distance between the initial numbers of two neighboring segments). To this progression of segments we add the (l+ 1)-st term l1i 1 l ', about whose type, of course, we know nothing. The segment l1i 1 l ' does not have to belong to the left halI of the
segment
tJ.. •
"1
15 any more, but must obviously belong to the segment
'1
\\Ie no .... carry over our construction, which we have executed up to now in only one of the segments .'\ l' congruently to all the other segments ,1\'i l O;;;il;;;l1. \\e thus obtain a set of segments \li2 O;;;il~l~ l;;;i 2 ;;;lj with two indices. It is clear that lwo arbitrary segments of this set with indices not exceeding l arc of the same type:
You no doubt see now that this process can be cont inued. We carry it out k times. The ('esults of our construction after the first slep were segments of length qk-l' after the second step, segments of length qk .. '1.' elc. After the k-th step, therefore, the results of the construction are segments of length go= I, i. e., simply numbers of our original segment Ll. Nevertheless we denote them as before by
1\.. " '1 2'" k
,~..
(2)
(I ;;;il' i 2 ,
. '" /'t;. . '. ,
- '1'2""s
....
ik;;;l').
.,
'1'2""s
We now make two remarks which are important for what follows. l) In (2), if s
tJ...
'1'2 ...
, . ' .' . . '., 5 's + 1 ••• 'k. -.~ - - il' 2'" 's ·s+ 1 ••• 'k
if I;;; i 1, ••• , is' i; ... ,i: ;;; l and I ;;; i s+l' i s+2' ... , i k ;;; l' (I ;;; s ;;; k). 2) For s '5:.k and i's = i s +1, tJ...1.. ••. -1.. and tJ...1 ••••. .1.s .' are obs
s
s
viously neighboring segments in the s-th step of our construction. Therefore for arbitrary indices i s+l' ••. , i k' the n urn bers t\. . .. . and tJ.. . . '. . appear in the same
·1···· s .1·s's+I····k
·1· .. •s -1·s·s+1····k
JI()f~ition in two such neighboring segments, so that (with i;=is+l)
16 ~. . .'. . '1""s-1 's 's+I"'\
(4)
a.
.
..
'1""s-1 's 's+I'" 'k
= d . s
§6 Now we are near our goal. We consider the following k + 1 numbers of the segment A:
ao =
a l ' l ' l ' ..• l '
a1 = 1\11'1' ..• I'
(5)
a2=
a11
1' ... I '
........ ak = a 1
1 1 ••• 1 •
Since the segment a has been divided into k classes, and we have k + 1 numbers in (5), there are two of these numbers which belong to the same class. Let these be the numbers a r and as (r< s), so that (6)
1\1 •.• 11' ...
1''''~1
'-v-' '-v-' k-r
••• 11'••• 1' .
--.--- '-v-' s k·s
We consider the l+ 1 numbers
(7)
ci =a 1 ...
1 i. .. i 1' ... 1' --.,,-' ' - v - ' - - ' r s-r k-s
The first 1 numbers of this group 0. e., those with i
so that
C,',O
= C.,,
C.
',s-r
'., ... ,. I ' •• , I ' '-v-''-v-' s-r-m Ie-s
= C&+1 ,and
hence
(O~m~s-r),
17
S
-r
C:+I-C.=~(C. -c.".m- i). l. m=l L,m I.
Because of (.J) C.t,m -
Thll~
c."m.. 1 =. \
\\C'
have
I ... I, · ,... , , r.. ••• l.. I ' ••• I ,- \ I ••• 1·' .' L • • • 1. I'••• I ' =,/+ l. ••• t "'r m • -.- _ _ '--..~ '-v-'" --...,.-..' , - ' - / m s·r-m ~-s m-l s-r-m+ik-s
the difference
find is indeed independent of i, \\hich completes the proof of ollr assertion. ) 011 ~pe how complicatf."d a completely elementary constructll)lI C.1fi ':iolTletimcs Le. And yel this is not an extreme cas(": in the 1I1'"\.1 chapter YOII ~ill ('ncoullt('r just as elementary a construction wilich is considerably more COIllI'IICJled. l3esides. it i" nnl out of the q11estion that van .lee \\df."rclf'n'" Ihf'tJrclIl aomits of all eVf'n <;impler proof, and all reseJrrh ill this direction can only be ~e1ellll"'!!.
CHAPTER II
Tin:
LANDAU-SCHNIREU1ANl\" HYPOTHtSIS AND MANN'S THEonEM
§1 You have perhaps heard of the remarkable theorem of Lagrange, that every natural nlLmber l.~ the sum of at most four sqluues. In other words, every natural number is either itself the squm·e of dnother number, or ebe the sum of two, or else of three, or else of four sllch squares. For the purpose at hand it is desirable to understand the content of this theorem in a somewhat different form. Let us write down the sequence of all perfect squares, beginning with zero: (5)
0, 1. 4, 9, 16, 25 .....
This is a certain sequence of ""hole numbers. We denote it by 5, and imagine four completely identical copies of it, \ . '\' 53'
5".
to be written down. Now we choose an arbitrary number a~from 51' an arbitrary number a~ from 52' an arbitrary number a~ from 53, and an arbitrary number a~ from 54' and add these numhers together. The resulting sum (*) can be 1) zero (if a 1 =a 2 =a3 =a4 =0); 2) the square of a natural number (if in some representation (*) of the number n three of the numbers aI' a 2 , a 3 , a" are zero and the fourth is not zero); 3) the sum of two squares of natural numbers (if in some representation (*) of the number n t\',o of the numbers up a 2, a 3 , a 4 are zero and the other two are not zero); 4) the sum of three squares of natural numbers (if in some representation (*) of the number n one of the numbers aI' a 2, a 3, a 4 IS equal to zero and the remaining three are not zero); 18
19 ~.I
till'
~lJm
of four squares of natural numbers (if in some repof the number n all four numbers are difTerent from zero). Thus the resulting number n is either zero or else a natural IIII,ul,.", which can he represented as the sum of at most four .. '111111.( .... , anrl it is clear thaL conversely every natural numLer can be ,,1.1 ;li,lt'oi hy the process which we have described. \OI\~ I('t us arrange all natural numbers n which can be obtained I,~ 1I11',IIlS of lIur process (i,e., by the addition of four numbers taken
,'· ... ·111 .. 1 iOln
""'111'1 lively from the sequences '\' 52' S3' .)4)' in orrler of mag1I,lud,·, in the sequence (
')
!\,Iwrp O
(1)
0, a l
(1) ( 1) ' a 2 ' ",,11m , .. ,'
(2)
0, a l
0, a
(k)
(2)
'
( 2)
a 2 ' ••• , am , .. "
, a
(k)
, ... , a
(k)
12m
, ...
\\r choose arbitrarily a single number from each sequence A (il 11 '.i'~k) anel add these k numbers toge-ther. The totality of all numhers ('onstructed in Ihis manner, if we order them according to magnitude, yields a new sequence (I)
... , n , ... m
IIf IIIl' same type, which we shall cull Lhe t!III"IH"'S ,j (1 I, A (2), , ••• A (k l:
sum of the given se-
k , A=A(I)+A(2)+ ... +A(k)= ~ A(') i'" 1
20 The content of Lagrange's theorem is that the sum S+S+S+S contains the entire sequence of natural numbers. Perhaps you have heard of the famou:s theorem of F!:'rmat, that the sum S + S contfUns all pnme numbers 14-hich leave a remainder of I when divided by 4 (i. e., the numbers 5, 13, 17, 29, ... ). Perhaps you also know that the famous Soviet scholar Ivan \fatveye"itch VinograJov proved the foJloVving remarkable theorem, on which many of the greatest mathcmaticians of th!:' preceding two centuries had worked without success: If u'e denote by r the sequence (P)
0,2,3, ,;,7,11,1:3,17,
consisting of zero and all prtme numbers, then the sum P +P +p contains all suUiciently large odd numbers. I have cited all these examples her!:' for only one very modest purpose: to familiarize you with the concept of the sum of sequences of numbers and to show how some clas:sical theorems of number theory can be formulated simply and conveniently with the aid of this concept. ~2
As you have undoubtedly observed, in all the exalllpies mentioned we arc concerned with showing that the slim of a rertain number of sequences represents a sequence which contains either completely or almost completely this or that class of numberrs (e. g., all the natural numbers, all sufficiently large odd numbers, and others of the same sort). In all other similar problems the purpose of the investigation is to prove that the sum of the given sequences of numbers represents a set of nllmbers which is in some sense "dense" in the sequence of natural numbers. It is often the case that this set contains the entire sequence of natural numbers (as we saw in our first example). The theorem of Lagrange says that the sum of the fOllr sequences S contains the whole sequence of natural numbers. ;'\jow it is customary to call the sequence A a basis (of the sequence of natural numbers) of order k if the sum of k identical sequences A contains all the natural numbers. The theorem of Lagrange then states that the sequence S of perfect squares is a
21 Lasis of order fOllr. ft was shown later that the scrrl1ellLe of perfect l:llbes fonns a basis of order nine. A little rf'llection shows that "\cry basis of order k is also a basis of order Ir + 1. In all these Hnd in mdny other examples Ihl' "density" of the SI1I11 \\hich is to be established is determined by particular prop('!'ti('s of the sequences thnt arc added togetht'r, i. e., by the spe(i.1I arithmetical natur(' of the numbers which go 10 make up these se'luences (these numbers being ejlht'r perfect sl]uares, or primes, Ill" others ()f a similar nature). Sixteen years u~o the distinguished Soviet scholdr I,ev (;enrichovitch Scltnirelmann first raised the 'lllf' ... tinn: To what extf'nt is the rlensity of the SUIll of several bCqllf'nc<'s .!('termined solely by the t!cn<;ity uf the summands, irl'f'"'pe('tive nf their arithmetical natllr('. lllis problem turned out to be nnt only deep and interesting, but also useful for the treatment of some c Ins,.,ical problems, During the intervf'ning fifteen years il rel:eived the attention of rnan) ()utstanding ,,;cholars, and it has ~iv('n rise to a rich literature. Before we ('an state problems in this field precisely dnd write til" word "dt'nsity" without quotation marks. it is ('vident that we mllst first agree on what number (or on ""hat numbC'rs) to use to lIleasure tht' "density" of our sequences with (just as in physics the \\ords "warm" dntl "cold" do 1101 acquire a preci!'e 'Scientific mpaning until .... C' havt' It'orned to meusure tempemture). A very convenient measure of Ihf' "density" of a sequence of numbt'rs, which is no .... used for all scientific problems of the kind we are considerin~. M1S proposed by L. G. Scltnirelmann. Let
be a seCJuence of numbers, whert'. as IIsllal, all the an are natural number!' aud an < an+! (n = L, 2, .•. ). \\ e denote by 4(n) the number of nalural numbers in the sequence (A) which do not exceed n (zero is 11111 counted), so that o~,f(n)~n, Then the ineCJuality
h"lds. J'he fraction A(n)/n, which for different n naturally has difft·rt·nt valuC's, can obviously be interpreted as a kind of average
22 density of the sequence (;1) in the segment from] to n of the sequence of natural numbers. Following the suggestion of Scbnirelmunn, the greatest lower bound of all values of this fraction IS called the density of the sequenc!' C·I) (in the entire sequence of natural numbcrs). \\e shall denote this density oy d(.·!). In order to become familiar at once ",ith the elementary properLi{'s of this concept, r recommend thaL you convince yourself of the validity of the following theorems: 1. (f a 1 > 1 (i. e., the sequence (A) does nnL contmn unity), then dCA) =0. 2. If an=l+r(n-I) (i.e .. the sequence ('0, beginning with ai' is an arithmetic progression with initial term I and difference r), then d(A)=l/r. 3. The densi ty of every geometric progression is equul to zero • .t.. The density of the sequence of perfect squares is equal to zero. 5. For the seqnenct' (/1) to contain th" {'nt ire "equellec of natural numbers (an =n, Tl = 1. 2 .... ), it is necessary and sufficient that d(:!)=l. 6. J[ aI::I) =0 and ;1 contains the number I, and if (>0 is arbitrary, then there exists a sulficiently large number m such that A(m)<wl. If you have proved all this, you are familiar I'nough with the cflflcepL of density to be abl", to use it. ]\0\\ r want to acquaint you with the pruof of the following remarkable, albeit very simple, lemma of Schnirelrnann:
d( 1+ n) ~ d("!) + d( B) -d(.,f) dun. The meanmg of this inp.qualily is clear: the den~ity of the sum of 1\\0 arbitf'dfY sequences of numbers is not smaller than the slim of their densi ties diminished by the product of these densities. This "Schnirclmann inequality" represents the first tool, still crude to be SlIre, for estimating the density of a sum from tllc densities of the summands. lIere is its proof. We nennte by ,l(TI) the !lumber of natural numbers ",hich arrear in the sequence ,I dnd do not exceed n, and by B(n) the analogous number for the seljuence B. For brevity we set dCI)=a, d(B)=/3. A+B=C, d(C)=y. The segment n,n) of
23 the sequence of natural numbers contains A(n) numbers of the sequence A, each of \\hich also appears in the sequence C. I.et a k and a k + 1 be two consecutive numbers of this group. Ilet\\et"n them there are a k + 1 -ak -] =l numbers which do not belong to A. These are the numbers
Some of them appear in C, e. g., all numbers of the form a,,+r, where r occurs in B (which \\1' abbreviate as follo\\s: r eR). There arc as many numbers of this last kind, however, as there arc numbers of B in th(' segment (I, 0, that is, B(l) of them. Consequently every segment of length I included between two consecutive numbers of the sequence -1 contains at least B(l) numbers which belong to C. It follows that the number, C(n), of numbers of the segment appearing in C is at leust
n, n}
A(T!) +I n(l)
where the summation IS extended over all segments \\hich are free of the numbers appearing in A. According to the definition of density, however, B(l)~{3I, so that C(71) ~ ·f(n) + PIl == A(n) + f3111 -A (71)1,
because "il is the sum of the lengths of all the segments which are free of the nllmbers appearing in 4, which is simply the number n-A(n) of numbt"rs of the segment (I, n) which do not occur in 1. But A(n) ~an, and hence
C(n) ~A (n)(I -(3) + (3n ~an(l-fJ) + {3n, which yields
C(n)jn ~ a+{3-a{3. Since this inequality holds for an arbitrary natural Dumber n, \\e have
y=d( (;) ~a +{3 -a{3. Schnirelmann's inequality (1) can be written
Q. E. D.
In
the equivalent
24
fonn I-d(A +Rhll-d(A)l1
-d(m!,
j
and in this form can easily be generalized to the case of an arbitrary number of summdnds: k
l-d(A 1 +A 2 + ... +A..):>; ft
-
n fl-d(.1.)I. i==l '
It is proved by a simple induction; you should have no trouble in carrying it out yourself. If we write the last inequality in the fonn k
d(A 1 +A 2 +···+A k );;;1-
(2)
}J 1Il-d(" i )l,
it again enables one to estimate the density of a sum from the densities of the summands. G. Schnirelmann derived a series of very remarkable results frolll his elementary inequality, and obtained above all the following important theorem:
r..
Every sequence of positive density is a of natural numbers.
baSIS
of the sequence
In other words, if a==d(A»O, then the sum of a sufficiently large number of sequences A contains the entire sequence of natural numbers. The proof of this theorem is so simple that I should like to tell you about it, even though this will divert us a bit from our immediate problem. Let us denote for brevity by A k the sum of k sequences, each of v.hich coincides with A. Then by virtue of inequality (2), d(·1 k);;; l-O-a)k. Since a>O, we have, for sufficiently large k, (3)
Now one can easily show that the sequence A2k contains the ",hole sequence of natural numbers. This IS a simple consequence of the following general proposition. LEM!'1:\.
If A(n)+B(rt»n-l.
then
n occurs
in A+B.
Indeed, if n appears in A or in 8, everything is proved. We may
25 therefore assume that n occurs in neither :f nor [;. Then A(n) = f( n -0 and B(n)= B(n·- 1), and consequently
l1(n- 0+8(n-]»,,-1.
°
Now let a p a 2 , •.• , a r and 1 ,1>2' ... , Os be the numbers tlf tLe segment (l,n-l) which appear in ·1 and H, respectivdy, so that r~ 1(n-l). s=Rlrl-l). Then all the numbers a l , a z'
... , ar'
n-b l , n-b 2 ,
... ,
n-b s
1(71 - 1) + R(n -1) of these IIl1mlJf'I"S, which is more than n-1. IIpncc one of the numbers III Ihl' lJl'pl'r ro\\ equals one of thc numbers in the lower ro\\. r .el '1,. -1/ -lJ k • Thl'n n=Cli+b k • i. e., rt appears in tI +B. HClllrning now to our objccti~e, w(' have, all the basis of (3), ror an arhitrary n:
(wifing t(l the l>ef,'ffient (1, n -1), There arc r+ s
=
ilnd tllf'refore
,\ccording to the lemma just proved, it follows Ihal n appears in
I k +A k =:l 2k • Hut n is an arbitr.u·y natural number. and hence our theorem is proved. This simple theorem led to a series of important applicdtions in the papers of [ .. C. Schnirellllann. For example, he was Ihe first to prove that the sequence P consisting of unity and all the prime numbers is a basis of the sequence of natural numbers. The sequence P, it is true, has densi ty zero, as Euler had already shown, so that the theorem which we just proved is not directly applicable to it. Uut Schnirelrnann \\fJS able to prove that P +1' has positive density. lIence p+p forms a basis, and therefore I' indeed also. From this it is easy to infer that an arhitrary natllrdl numher, with the except ion of J, can, for sufriciently large k, be represented as the sum of fit most Ir primcs. For thaI lime (1.930) this result was fundamental and evoked the greatrst intC'rest in the scientific \\orld, At present, thanks to the remarkable \Iork of I. \1. \"inograd-
26 ov, we know considerably more in this (Iireclilln. us I already related to you at the beginning of this chapt er.
In the preceding it was my purpose to introdllce )011 in the shortest way possible to the problems of this singular ana fascin<.lting branch of numbcr theory, whose ... tudy began \\itlt I,. G. ~chnirellJlann's remarkable work. The immediate goal of the present chapter. hov.evcr, is a specilic problem in this field, and 1 now proceed to its forrnulatinn. In the fall of 19:3], upon his return frllm a foreigll tonr, L. G. Schnirelmann reported to !HI IllS convers.tli"n!'> v.ith l.andau III COILingen, ano related among other things that in the course of these conversations they had discoHre(1 the following interesting fact: In all the concrete pXHlllple!o> that they were ahle to devise, it was possible to replace the inequality d(A +R)~d(A)+d(H)-d( t)d(/n,
which we derived in *2, by the sharper (.rnd simplPr) int·quality
That is, thl' density of thc sum alvyays turueo flul to be at least us large as the sum of the densities of the sllmrnanrls {under Ih£> assllmpt ion, of course, that d(,.1) +d(B) ~ I). Thcy thereforp naturally assumed that inequality ('1) was the expression or a universal law, bllt the first atlempt& to pruvc this conjecturc \\pre Ilnsllcces~ful. It soon became t'vid"nt that if their c()nJecture WiIS correct, the road to its proof would bc quitc dilTiculL. \\e "ish to note at this point that if the hypothetical inequality (4) does represPllt " univprsal law, then this law can be gencralizeo immcdiately by induction to thc cuse of an arbitrary numbcr of summands: i. ('., IIn(ler the assumption that k
lJ(n~l
i=1
,-
Voe have k
(5)
k
d( ~ Ai)~ 2 d (-Ii). i=1
i=1
27 This problcm coule! not help hut attract the attention of scholars, bN'ause of the simplicity and elegance of the p;eneral hypothetical IlIw 0) on the one hand, and nn the other becallse of the sharp conIrllst bet¥.een the elementary character of the problem and the diflieulty of its sollltion which became apparent already after the first i1l1acks, I myself was fascinated by it at the time, and neglected 1111 my other researehcs on its uccount. ~:arly in 19:12, aILer several IIIllnths of hard ¥'ork, l succeeded in proving inequality (4) for the luost important special case, d(A) = dW) (this case must be considNetl as the m()st important becalls!' in th". majority of concrete proLlpllIs all the summands are the same). At the same time l also I'rnvcJ the ~cneral inequality (5) under the assumption that d(A 1)= d(:l z )= ... =d(A,J (it is eus)' to see that this result cannot bederivI'c! from the preceding onc simply by induction, but requires a spel'ial proof). The method which I used was completcly elementary, but v('ry complicatNl. I "US late-r able to simplify the proof somewhat. Be that as it may. it was bllt a special case. For a long time it scemed to me that a 1I0n(' too subtle improvemcnt of Illy method should Ipac! tll a full solution of the problem, but all my efforts in this direction prov('d fruitless. ! In the meantime the publication of my work bad all~'arted the utt~lItion of a wide circle of scholars in all countries to Ihi! I andallSchnirclmann hypothesis. \Iany insignificant results were obtained, and a whole litl.'rature spran~ lip. Some authors carried over the problem from the domain of natural numbers to other fields. In short, the problem became "fashionable". r,earned societies olTered prizes for its solution. My friends in Englanl"! wrote me in 1935 that a ~ood half of the English mathematicians had postponed their usual work in order to try to solve this problem. Landau, in his tract devoted to the latest .ldvance<; in additive number theory, wrote that he "should like to ur~e this problem on the reader". But it proved to be obstinate, und withstood th". efforts of the most able scholars for a whol e series of vear~. It \'oilS not IInti I 1942 that the young American mathematician \Ianll finally oisposed of il:1 he found a complete proof of inequality (~) (alld hence also ofinequality (:))). His method is wholly elemt'ntar) dnd is related to my work in
28 form, although it is Lased on an l"ntirdy different idea. The proof Iq long and very complicated, and I could not bring myself In pr('sent it to you here. A year later, however. in 1943, Artin an.l Scherk published a new proof of the same theorem, which rests on an altogether dift'erenl idea. It is considerahly shorter and more Iran~par{'nt. though still quite elementary. This is the proof that I shnul'] like to tell you about: 1 have written this chapter on its account, and it forms the content of all the succeeding s('ctions.
H Suppose then that A and B are two sequences. We set .1 + U = I.. Let 4(11), d(A), etc. have their usual meaninR' \\e rt'call that all our sequences hep;iu with Zl"ro, but that only th(' natural nWllb('rs appearing in these sequences are considered when calculatin~ Atn), R(n), C(n). \\'e have to prove that d(C) ~ d(A) + d(B)
(6)
provided that d(,t)+d(R)~ 1. For brevity we set d( 1)=a, d(R)={J in what follows. FUNDAMENTAL LEMMA. If n is an arbttrary natllral number, there exists an integer m (l ~ m ~ n) silch that
C(n) -C(n - m)~ (a +f3) m.
In othl"r words, there exists a "remainder" (n -m + 1. n) of the segment 0, n). in which the average density of the sequence C is at least a +{J. We are now faced with tYto problems: first, to prove the fundamental lemma, and second. to sho\\ that inequality (0) follows frolll the fundamental lemma. The second of these problems is incomparAbly easier than the first, anJ we shaH therefore bl'~in witl! the second problem. Suppose then that the fundamental lemma has ulready been proved. This means that in a certain "remainder" (n-m+ 1,,,) of the segment (l, n) the average density of the sequence I. is at least a + {J. By the fundamental lemma, however, the segment (l, n -m) again has a certain "remainder" In-m-m'+l,n-m) in which the average density of C is at least a+fJ. It is clear that Ly continuing
this process, the segment (l, n) is eventually divided into a finite number of suhsegrnents. in eHch of which the average density of C is at least a+f3. Therefore the avcrage density of C is also al least a+~ in the ",hole segment (l,n). Since n was arbitrary, however, we
!.u....... I).
~;.
D.
Thus the problem is no", reduced to proving the fundamental lemma. We now turn to this proof, which is long anr! complicated. ~:; {I; ()
H \1 .\ L
:; E 0 Ij r,:
{I;
C ES
In all that follo",~ we r:;hall rc~ard the numbcr n as fixed, and all sequences which we investigate will consist of the numher 0 and certai n numbers of the segllll"nt 0, n). \\ I.' af.,>Tee to cull such a sequence ,...j normal, if it posses!:>f"S the follo"'ing property: If thc arbitntry numbers f and f' of the segment (1. n) do not appl'ar in {Ii. then neither does the numher f+ -T/ appear in N (",herc the case f= is not excl uded). J£ the number n uclongs to thc sequence C. then
r
r
so thut the fundamental lemma is tri\ially correct (m=]). C()nscquently wc shall assume in the scqucl-I beg you to keep thiS In mind-that n docs not occur in C. To begin with, the fundamental lemma is easy to prove in case the sequence C is normal. Jndeed, let us denote by m the smallest positive numher which docs not appear in C (m ~ n because n, by asswnption, does not occur in C). I.et s hc an arhitrary integer lying hetween n-m and n; n-m<s
C(n)-C(n-m) =m-l.
30 On the other hand, by the lemma on p. 24, since m does not occur in C=A+B we have A(m)+B(m);;;;m-l. Consequently
C(n) - C(n -m) ~ A (m) + B(m) ~ (a + (1)m, whieh a/(din proves th!:' valiclity of the fundamental lemma.
§6 C.\ t'\ ONICA L
~~).
TEN SIO'lS
\\e now turn our atll"ntion to the Cdse where the sequl'lll'C C = I i- B is not normal. In this case we shall add to the set R, according to a very de6nit!:' srheme, numbers v.hich it due .. Bllt (;ontain, and thereby puss From B to dll e'(tenoeo set J ' I'ltl' set .-1+Rl =C 1 evidently will then be a certain extension of tllf~ s!'t L. ,-\s I saio before, this extension of thE' SE'ts B dIld r: (the <;cl ,r remains unaltered) will be .lelined precisely and unambiguously; it is possible if and only if the set C is not normal. We shall call this extension
n
c¢C. (' ltC, c +c '-n EC. Since C = ., + B, it follows thot
(8)
(aE
c+c'-n=a+b
I, bEN).
Let {30 be the smallest numher of thp set B which rnn play thp role of the number b in equation (8). In other .... ords, {30 is the small('sl integer bE which sati!ifies equation (8) for suitably rhos('n lIumbers etC, e '¢C, aEA of the segment (0, n). 'fhi!! numbel' (30 will be call'Cd the basis of our extension. Thus the equation
n
(9)
e+e l -n=a+{3o
necessarily has solutions in the numbers c, ditions
e:
II
satisfying thE" con-
31 c ¢ c, c' ¢ C, a E A, where all three numhers belong to the segment (0, ,,). \\ e write all numbers c and c' which .. atisfy equation (9) and the enumerated conditions, to form a sct C*. Clearly the sets C and C* do not have a single element in common. We call their union* (i. e., the totality of all numhers which uccur either in C or in C*)
the canonical eXlension of the set C. r,et us now examinl' lhl' ('x pression f3 0 + n-c. If c here is allowed to run through .tIl the nUrtluers uf the set C* Just constnJC'tf'd, the values of this expres!'.itln form a certain set If". :\ccording to equation (9), every suth number f3o+n-c (CEC*) can be written in the form c'-a, \~here e'EC"', aE ·1. Lei b· he an arbitfiu'y number oee urring in R*. Si nee it is of the form {3o+n-c, it is ~f3o~O:
IlUn*=B 1 and call the set B 1 a canon iCIl I extension of the set B. Let us show that
:I-fR 1 =C 1 · First, let aEA, b 1 EB 1 • \\e shall prove that a+b 1 Er. 1 • Fmm b 1 EH 1 it follows that either blEB or b 1 EfJ*. T£ blr:.N, Ilwn (11-/)1 EA + H =CcC l ' If b1 EB*, however, then a +b 1 either ueellr~ in C, and hence also in Cit or a + b 1 ¢ C. Uut in this case (since bl> as an element of the set B*, is of the form f3o+n-c: c'¢e) WI' ohtain
Therefore ·Here and in the sequel we use the symbol are using the sytl1bol + in another sense.
Uto
denote the union of seLs, since ....e
32 c+c'-n=a+{3oEA + n = L. where c set
¢C
and c' ¢
r:.
Hut thpn ,Icrordi II).! tu the dcfini tion of the
C"'. 0.1' .. II .
Thus wp ha,,!' sh,mn that 1 -t 8
1 ( ( : 1.
Tu prove the imerse relation. let liS assllme thut r E('l, "llIdl ntc,m!-l thut pithPr c EC or c EC*. If c EC, tiwn c=u+b, a 1"
eli l ' If.
1. bEN
ho\~e\'er, c EC1, then, for a certain a E,I, the nllTliLer
G-a. "'" \\e knnw, "l:cur!-l in U*. \\c have c=a+b* E I +/{* - t Th('refurc C 1 C.-I + U l' \\ e al so pruved above that A + H 1 :-.e'jlJently (;1 = ., + 8 1 , No\\ r("call th'lt according to 01lr .Jssumrtion,
7/
¢c. It
r: l' i:-.
b*=
,-n 1 • l. "n-
I'H<;)
to
see-allJ this i" illlportant-that the numoer n due., lIul appcar in the extension C 1. For if \\e had n EC*,
\\P
c01l1rl, by the definition of
C"', put c'=n in equation (9), which would yipld C~1l1 {30 E I +-B=C, whereas c ¢C according to (9).
If the extended sequence C\ is not yet normal, then, beea"s(> of '1+Rl=C j and n¢C 1 , thf' sets I. N 1 , and C 1 form a triple \\ith all the properties of the triple A, n. C that arc n('('c-;smy for it new eanunical extension. \\e taI..e a ne\\ Lasi~ (31 "E thi., extension, definp the c('mph'mf'ntar), s!'ts
H!,
C1
a<;
before, pill
·i(,'~. It is pvidf'nt that thi,,; proccss can be c'lntinlJerl until one !If th,; (,,,tensIOns C h proves to be normal. Obviously thl'5 ('<1<;(' mu<,t certainly amI are <.Lle to a!-lsert onre more that ,'+8:;=C 2 ,mol
f/
take place, because in cvery extension \\c add nc" number::. Lo the spts nil and Gil withont o1!erstepping the bounJs of the segment (0. n). In this way we obtain the finite sequences of set--
IJ = B 0 c B 1 C .•• C Bh ' C = Co C C 1 C ••• CCh ' "here every Bp.+l (respectively GIl+l) contains nurnb('ro.; "hich do not upped!" in Bil (C Il ) and which go to make up the set R~ (C~), so
33
that
BP.+1 =Bp. U B~,
CP.+1 =Cp. UCt
(O~p.~h- 1).
\\{' denote by f3,l the basis of the extension which cnrries (BIl' \.p.) into (B ,Hl • CP.+I)' \\c have
Finally, the set C h is normal, whereas the ~ets Cp. (O~p.~h-l) are not. ~7
PROP}.ItTIES OF THB C\:'lO"lICAl. EXTENSIO:'lS
\\ e shall now fo rill uI ale and prnve in the form of three icmmas those propcrticg of the canonical extensions \\hich are ncedt',j lalcr. Onl} r.emma 3 wdl havc further application: I.cmmas 1 and 2 are requ ired <;<0 lei y for the proof of I .emma 3. I .~: \1 '1·\ 1. (1p.) f:3P.-1 (l ~ P. ~ h -1); l. e., the ba<:es of successive canonical extensions {('rm a monotonically increa<;mg sequence. In fact, sinC"l' {:3p. e8J! = BP._1 LiLJp_l' either {:1p. eBt-1 or {:1p. e LJP._I' If f3p. eBt_l' then f3p. i!:o of the form
f3p. =f3P.-1 +n-c, whpre CEC~_I( ep' an,] thf'ff'fllre c
{3p.-1' Hnd Lemma 1 is proved. If f3p. EBp._I' hO\~('ver, Lhpn b} the definition of the nlllllbcr f3p. there exist integers a e ·1. c ¢ CIl' c'r!. ep' such thut
c + c '-n=a+{:1p. I'; Cp.' But for {:1p. E B P.-l' we have (10)
where c¢Cp._l' C'¢CP._l' lienee, because of the minimul property of flp.-l' f3P."?,{:1P.-I· If {3p. = {3p.-1' it would follow from (10) and the definition of the set C 1 that
P_
CI';C~_I( CfL'
C'I';Ct_ICC,l'
Uoth are false, however, and thf'ff'fore {3p. >{3P.-I' In the sequel \\c shall den(.te by m the smallest positive integer
34
which does not appear in C h' LEMMA 2. If ceCt (O;;;/l;;;h-l) and n-mn-m+ f3/l' That is, all numbers c of the set C~ u'hich lie in the £/Iterva{ n - m < c < n are embedded in that part of this segment which is characterized by the inequalities n - m + f3/l < c < n.
We have to show that
c+m-n>f3/l'
It follows from n -m < c < n that O<m+c-n<m. Therefore, by the definition of the number m,
m+c-nEC h· Now
We consider two cases. 1) If m+c-n ECW then
m+c-n=a+b/l' Out m ¢ C /l and c ¢C/l (the latter because c E C~). Therefore because of the minimal property of f3/l y,e must have b/l~f31l' b/l=f3/l it would folIo\'< from the definition of the set C~ that m E C~, which is false because C~CC/l+ICCh and m¢C h · COlJsequently b/l>f3 W so that
rr
and I,emma 2 is proved. 2) If c'=m+c-n EC~ (/l;;;v;;;h-l), then, by the definition of the set C~, c' satisfies an equation of the form (9),
c , -a= f3 v+n-c " , where aEA, C"EC~. Hence c'~c'-a>f3v~f3/l (where the last equality is given by Lemma 1), and r,emma 2 is again proved.
'-
In-
35
T.E ~I MA 3. IFe have Cit(n) -Cit(n-m) = Bit(m-I)
(O~p.~h -1).
Tltat is, the nu.mber of integers CECit in the segment n-m
J .eL
liS
examine the relation
01 ) By th~ vcry d('finition of the sets U~ and Cit' C ECit implies b EBit, an,) conv('rsc\y. If, in addition, n-m+{3p.
Cit(n) - C~(n -m + (3p.) = B~(m -1) - Bp(f3p.). Oy Lemma 2, Cit (n-m+f3p.)=C p(n-m). On the othr hand, every bE Bit cun be expressed in the form (11), where c
Q. E. D.
§8 PIlOOF OF TT-fTo: FUNDAM~NTAL LDIMA
It is very easy nnw to prove the fundamental lemma by proceeding from the results in §5 and appealing to Lemma 3 which was just proved. If we apply the result of *5 in tbe form of inequality (7) to the sequences A, B h' and C h (which is permissible because of the normality of Ch ), \'I.e find that (12)
Ch(n)-Ch(n-m)~A(m)+Bh(m),
where m is the smallest positive integer which does noL occur in (,h' Obviously m ¢A and m ¢B h , so that we may write A(m-l) and Bh(m -1) instead of 4(m) and Bh(m), respectively. \\e have Ch =CUC* uq LI ••• UCA_l'
Bh=B uB* UBi J ... UBt_l'
36 where the sets appearing ally exclusive, so that
In
anyone of these tv-.o un IOn,,; ure mutuh-l
Ch(n)-Ch(n-m )=C(n)-C(n-m)+ ~ I Cr(n)-q/n-m) h-l 11=0 Bh(m)=Rh(m-I)=B(m-l)+ ~ U~(m-l); 11=0
b
-y.,e have of course put C = C*, that
Bb = B*.
I,
On ac COllnt of (12) it follows
h-l
C(n)-C(n-m) + ~ I C~(n)-C~(n-m) 11=0
I
h-l ~.f(m)+H(m-I)+ ~ B~(m-1).
p.=o By Lemma 3, however, C~(n) -C'/t(n-m) =B~(m-l)
(O~I1~h-l) ,
so that the preceding inequality becomes C~Il) - C(n - m) ~ A (m) + H(m - 1 ) = A (m) + B(m) ~ (a + f3) m,
whi ch proves the fundamental lemma. As we saw in ~ Il, this also completes the proof of :'Ilann's theorem which solves the fundamental metric problem of additive number theory. Doesn't Artin and Scherk.'s constrllction have the ~tHmp of a magnificent masterpiece? I find the outstanding combination of structural finesse and the extremely elementary form of the method especially attrllctive.
CIIAPTF.H
Ai\
1,:u:~IF;l\I'rAItY
111
SOI.UTIO;'II OF
WAHI~G'S PHOBL~\1
§1 \ou will recall the theorem of Lagrange, which \\11S r1i!'cm;sed at the be~inning of the prt"ceding chdptl'r. It says that c"l'ry natural number can be express('ll as the slim oC at m05t four !'f]uures. r also showed you that this lheorem could be stalerl in enlirely different terms: H four sequences, each identical \\ith
0, ]~, ~2, ... , k 2 ,
••• ,
are added together, the r('sulting sequ('nce containe; all the natural nWllLers. Or even lIIore Lrit"fiy. the sequt"nce (A 2) is a bnsis (of the !'equence of natural nUllIb('rs) of order four. I also llll"ntiol1('d that. dS hurl been 'Shown later. the "(''fuen CPo
waR
II
l.f
cubes
basis of orriN nin.'. ,\11 tht"se facts lead in u natural manner
to tltt" h}1l1lthesi s that, for dn urbi trary naturul numbl'r n, the sequence
IS a basis (whose order of course depends on TZ). This conjecture was also aelually propounded by \\arinp; as early U8 the eighteentll cenlury. The problem proved to be very difficult. however, and it was not until the beginning of the present century that the universul validity of\\aring's hypothesis \\as demon8tratcd, by IlIlLert (1909). Hilbert's proof is not only ponnerolls in its form ,I I aspt"ct and based on compl icuted analytical theories (multi pIc IIltegrals), Lul also lacks transpllrency in {'orH:cptlZal respects. The eminent French mathematician Poincare \\rote in hi~ suney of IIilhert's creative
38 scientific work, that once the basic motivations bphind this proof were uuderstooo. arithmetical results of great importance would prubably flow forlh as from d cornucopia. In a certain sensp he \\<JS right. Ten to fifteen years later, new proofs of Hilbert's theorem were furnished by Ilardy und l.ittlewood in ~~ngland find by 1. \1. Vinograuov in the l"SSH. These proofs were again analytic ami formally un\\icldy. but diITered favorably from Hilbert's proof in '[heir clarity of method dIld their conceptual simplicity. which left nothing to be desired. In fflct, because of this, both methods became mighty scourees of nev. arithmetical theorpms. But when our scienep is concerned with sHch a t'omplpt('l y clemen tary prohl em as Wming' s probl em, it invari abl y nltcrnpts tn lind a solution "hicl! requires no concepts or methods transcpn,ling the the limits of elementary ariLllnletie. The search for sllch an elementary proof of II aring's hypothesis is the third problem which I should like to tell YOIl about. ~lI('h u fully ('Iernentary prnof ()f lIilLert's theorem was lirst oLtained in 194-2. by the young :'-l
39 proof that the sum of a sulTicicntly large number of sequences (A n) is a sequence of positive density. As soon as this is accomplished, we can, by virtue of the same general theorem of Schnirelmann, regard Hilbert's theorem as proved.
§2 TJI~:
FUNDA"IENTAL
LEMMA
[f y,e add to~ether k sequences, identic(ll with ,tn' acconlin~ to the rule in Chapter If, we evidently obtain a sequcnc-e ,,~k) y,hich
contains zero and all those natural numbers which cun be expressed us U Slllll of at most k sUllllndrl(ls (If the form xm, \\ here x is an arbitrary nalural numh('r. In other words, the number rn belongs to the sequenl"C A~k), if the cquation
(])
n
n
n
X1 +X2 + ••• +xk
=
m
is solvable in nonne~ativt' integers Xi (l ~i~Ir). ·\s We' saw in § 1, the problem is to show that, for slllfil-ientiy large k, the sequpnce .,~/r) hns it positivI! dl'n"ity. For preassi~ed k and m, equation (1) in generdl Cdll he solved in sevcnJ different ways. In the sequel \\e shall dcnote hy rk(rn) the number of these \~ay", i. e., the number of systems of nonnegative integers X1>X2t •••• xk \\llIch satisfy e'lllation (I). It is clear that the nlllllher In oc-curs in ,'~A) if and only if rk(rn»O. In the following. we shall u:"snme the numLer n to he given and fixed, and shall therefore cdll all numbers which depcnd only lIpon n, constants. Such constant:" \\ill he denoted by the (plter c I)r c(n). \\here such a constant c may have differenl values in different parts of our discussion, provided merely that these values are constants. Perhaps you are rather unused to sllch "freedom" of nolat ion, hut you wi II soon become familiar with it. It has proved to he very convenient, and appears more and more frequently in modern resf'urch. F'UND-\MENTA L LEMMA. 1here exist a natural number k=lc(n), depending only on n, and a constant c, such that. for an arbitrary natural number N,
(2) Once more, as in the prect'tiing chapter, we arc faced y,ith two
40 problems: first, to prove the fundamental lemma, and second, to draw from the fundamental lemma the conclusion that we need, viz., that the sequence A~kl has a positive density. This time again the second problem is considerably easier than the first, and we shall therefore begin with the second problem. It follows immediately from the definition of the number 'k(m), that the sum
represenLs the number of systems integers for v. hich
(X1, X 2 , ••• ,
x k ) of k nonne~ative
(3) E"ery group of numbers for which
Os.x. - ,-s.CV/k)l/n obviously satisfies this con.lition. To sdtisfy these ineqllalities, every Xi can evidently be chosen in more than (N/k)l/n diffel'ent ways (xi=O,I, .... [(A/lc)l/n]).* :\fter an arbitrary choice of this sort, the numbers Xl, X2' ... , X k may be combined, and so we have more than (N/lc)kln dilIerent possibilities for choosing the complete system of integers shows that
Xi
(l ~i ~k) so as to satisfy condition (3). This
(4)
\\e assume that the funrlamental lemma has been shown to be correct, nod that inequality (2) is satisfied for an arbitrary N. We now have to verify that inequality (2) is consistent with inequality (4) whi ch we proved, only if the sequence A ~ k l has a positive density. The idea behind the following deduction is very simple: In the sum Rk(iV), only those summands 'k(m) are different from 7.ero, for which m occurs in A~ kl. If A~ kl had density zero, then for Luge .'\ the number of such summands would be relatively small: because of (2), however, every summand cannot be very lar/Ze. Their sum Rk(N), therefore, would also be relatively small, whereas according to (4) it must be ruther large. • [a] denoles the large~t integer:::;: a.
41
It remains to carry out the calculations. Suppose that d(A ~ k »=D. Then, for an arhitrary small (>0 and a suitahly chosen ,V,
Here the number ,\' may be assumed Lo be arbitrarily large, because A~k) (for an arbitrary k) contains the integer 1 (bear in mind Problem 6 on p.22, which you solved). Applying the estimate (2) we get n
Rk(\)=
~ rk(m)=rk(O)+ m=O
N
~ rk(m)
and hence, for sufficiently large J\,
R k C\ ) < 2Cf,,\'k; n. For slJlliciently small
E,
2c« (I; k)kln, so that
which contradicts (4), Therefore we must have
d(A(k» >0. n But, as \\e alrf'ady kno\\. this proves lIilbert's theorem. 'ou see how simply it all cornes t~ut. But we still have to prove the fundamental lemma, and to du this we shall have to travel a long and din'jeult road, as in the preceding chapter.
§3 LEMMAS
CONCERNDIG
LlJliEAR
EQI,ATIO""S
\\e shall have to go far back. It will therefore be well for you to forget completely for a while the problem which has been posf'd. r shall call your attention to it when we return Lo It later. !light now, ho\',ever, \',e have to find some estimatt"s fo/' the number of solutions of systems of linear equations, The lemmas of this paragraph, moreover, are perhaps also of intrin"ic interest. independent of the problem for" hose sol ntion they are required here.
LE \1 \J:\ 1. In the C'7ua1i on
42
(5) let ai' a2, m be integers with la21 ~lall ~A, and let al and a2 be relatil'ely prime. Then the number of solutions of eqlwtioTl (5) satisfying the inequalities IZ11~A, Iz:;>I~A, does not exceed 3.t/la11. Proof: We may assume that a i >O, because otherwise we huve merely to replace Zl by - Z l in every solution. Let IZ1> z 2 1 ane! 1z{, z;1 be two different solutions of equation (5). Then from alz1
+a2z2=m,
atd -l-a:,zb =m we get
by subtraction. I\ccordill/?:ly the left-hand side of this equation must be divisible by ai' But* (ai' a2)= 1, and consequently Z;'-Z2 mllst be divisible by a1' 'Jow Z~';'Z2' and therefore IZ;-Z21, as a multiple of al, is not smaller than a1. Thus, for two distinct solutions Iz 1, z) and lz~, z;1 of equation (5), v.e must have IZ;'-z21~a1' In every solution lz l' zd of equation (5), let us agree 10 call Z1 the first member tim! Z2 the sec(lnd. [t is obvious thai the llumber of solutions [)C equation (S) which satisfy the conditions IZ11 ~A, IZ21 ~A. is not more than the numLer t of second members which occur in the interval <-·1, A>. Since we have proved that two such second members are at least the distance al apart, the difference between the largest and smal\t'st second members occurring in the interval <-II, A> is at least a 1 (t- 1). On the other hand, this difference does not exceed 2.1, so that a1(t-l):?;2·/'
(t-l) ~2A/a1' t~(24/ai)+ 1~~4/a1
(hecause, by assumption, a 1~A. and therefore 1 ~A/(t1)' This proves Lemma 1.
LE MM A 2. In the equation (6) • (al. a:;»
denotes the greatest Common divisor of the integers a1 Bnd a:;>.
43
let the a, and m be integers satisfying tlte conditions'"
Then thp number of solutions of equation (6) satisf""ing the i"pqualities IZil~A (l~l~l). does not exceed
ell) 11- 1 /11, u,here II is the largest of the numbers la11.la21 • •.. , a constant depending only on l.
lall,
and c(l) is
Proof: If l=2, I.emma 2 obviously becomes Lemma 1 (with c(2)= 3). Accordingly Lemllla.2 is alrcully vt>rifierl for l=2_ We shull therefore assume that 1~3 and that the truth of Lemma 2 has already been established for the case of l-I unknowns. Since the numbering is nnimportant, we may assume that lall is the largest of the numbers la1i, IU21 • ... , la,l, i. e., fI = lall. There arE' 1\"'0 cases to com,i.\cr. I) Ul =a?=... =a[.l=O.Since (a1>(l;?1 ... ,tl/)=I, ""t' hdve la l l=lI= 1, sn that the ~iv('n eqnation is of the form ±;:l=m. In this equation euch of the unknowns ;:l';:, .... ,z/_l can obviollsly assume an arbitrru'Y illte~ral valuc ill the interval <-il. :1>. and hencc at 1II0st 2:J +] ~3A values all toln. As for zl' ho""ever, it can assume at 1II0st one value. Conseqllentl) the nwnLer of solutions of the given equation satisfying the ine'lllUlities IZil~4 (1~i~I), rlOI'S nnl exceed
which proves Lemma 2 for this case. 2) If at least one of the numbers zero, then
a1, a 2 , ... , a l _ l
is dilTerent from
(U\.CL 2 , .. ··a l _1 )=o
exists. Let us denote hy II' the lar~est of the number~
Suppose now that the numhers ;:1, Z2> ... , zl salisfy the p;iven equation (6) and the inequalities IZil~A (1~i~l). \\'e set *(a1'~'
•••• al) denotes tht'
grpatest
common
44 and hence
Then obviously (~
and 1-\
lom'l~ i~ laillzil~ioll'A, y,hi ch impl iE'S that
Im'l ~lfI '..1_ Thus, if the numbers Z1, Z2' ••• , zl satisfy equation (6) and the inequalities Izil ~A (l~i~l). then the integer m' exists, which, with these numbers, satisfies equations (7) and (8), where 1m 1~lll';I. But in equation clearly 8~ lall and (8, Il t )= 1 (othery,isc we should hav(' (a1, a 2, .. _, a t _!, at» I). ilence, by Lemma 1, the number of solutions of equation (8) (in the unknoy,ns m', zl)' for which Im'l ~ 1II~'1, IZII~A
un
tions
IZ1' Z'2' .... zil
of equation (6) which satisfy the inequalities
IZil~A (l~i~l), does not exceed
(3L11 'A/
!at I) c(l) .1 1- 2/11 ' = c(L).,j/·l /Iail =.
dl):ll-l/ II,
which completes Lhe proof of Lemma 2. * "'e shall now investigate the totality of equations of the form (9)
where lail~A (l~i~l) and, as always. all a i are integers. Let B he a posi tive number whose relation to the number A is described by the inequalities l~A~B~c(l)AI-l. and let l>2. We now want to estimate ·You have probably noticed that in the last chain of pquations the .. ymbol c(l) occurred in different places with different meanings. On p. 39 \ prepared you for such a us<> of this symbol.
45 the SIDn of the numb('rs of solutions zi' IZil~B (l~i~l) of all the equations (9) of this family. 1~ Fi rst let us make a sl'parate examination of equation (9) for
ul=a2="'=U l =0 (it is a member of our family) and estimate the number of its solutions which satisfy the inequalities IZil~;Fl (l~ i~ Our equation is obviously satisfied by an arbitrary system of numbers Zl, Z2' ••• , zl' and we have merely to calculate how muny such systems exist which satisfy the inequalities IZ11 ~R, IZ21 ~R, ... ,lzll~B" Since the interval <-B,+B> contains at most ~B+1 integers, each zi can assume at most 2B + 1 different val ues. Consequently the number of systems IZ1' Z2, •• "' Ztl of the type in which we are interested does not exceed (28 + 1)1 ~ (3R)1 =c(l)Rl. By our hypoth('sis, however, R~c(l)Al"l, so that c(l)8l=c(l)Rl"18~c(l)(1R)l-.1
n.
lIence. for the case y,here Ul =a 2 = ••• =al=O. equation (9) has at most c(l)(A R)l-1 solutions of th(' type we are interested in. 2~ Even jf only one of the c(lefficients a i is dilTerent from zero, the greatest common divisor of these coefficients, (Ill. a~l> """' a l )=8, exists. Suppose first that {) = I, and Ie t lf be the I ar~est of the numbers lail (i = 1.2 • .... l). r.ll'dfly II is one of the integcrs in the interval <1,.4>. Ilencc. II is eitller bety,een A and A/2, or between ,1/2 and Al~, or between ..1/4 and ,I/R, etc. It is therefore possible to find an integer m ~ 0 such that (10) According to Lemma 2, for an equation of the form (9) in which
8=1 and /I satisfies the inequalities (10), the number of solutions zi' IZil~R, does not exceed
On the other hand, it follows from (10) that
(ll) Consequently the number of equations of type (9) for which the inequali ties (10) are satisfied is at most equal to the number of equations of the same type which satisfy the conditions (ll), i. e.,atmost
46 Thus the sum of the numbers of solutions IZ,I ~B of all such equations of type (9) for which 0=1 and A2-(m+l)
Summing this estimate over all m~O, we reach the following conclusion: The sum of the numbers of solutions Izil'i"R of all equations (9) for which lail~A (l~i~l) ando=l is at most e(l) (A B)I-l.
3':' It remains for us to figure Ollt the numbers of sol utions of the required type for equations with 0> 1. In this case equation (9) is evidently synonymous with the equation
where only
and the number 4 has to be replaced by the number A/o. As we saw in 2°, the sum of the numbers of solutions 1z.l ~ B of all such equations, for a given, fixed 0, does not exceed* e(l) (Ao-I.B)I-l = e(l)(A B)I-15-(l-l).
Clearly now we have merely to sum this expression over all the possible values of 0 (l'i"o'i"A). Thus we find that the sum of the numbers of required solutions of all equations of the form (9), where lail'i"A. (l'i"i'i"l) and not all ai are equal to zero, does not exceed the value A
eel) (AB) l-l
I o-(l-l) < c(l) (AB)l-l.l-1-=e(l)(AB)l-1. 0=1 l-2
[To obtain the first relation we employ the inequality A
I (l/nq+l )«q+ l)/q, n=l *Since instead of A we now have to take the smaller number AIO. it is conceivable that the assumed condition B'i"c(l)A.l-1 is violated_ You can verify, however, without any trouble, that we made no use of this assumption in Case 2°, and that the result in
'fJ therefore
does not depend on it.
47 which is valid for an arbitrary natural number q and for an arbitrary 4 ~1 (we denote by q the number l-2, which is positive hecause we assumed that l> 2). Here is a simple proof: For n~ 1 we have n- q - {n + l)-q =I(n+ l)q -nql/ nq(n+ l)q =(nq+qnq-l+ ••• +l-nq)!nq(n+ 1)q ~qnq-l/nq(n+ l)q > q/(n+ l)q+l, and hence
By substituting successively n=1,2, ..• ,-1-1 in this inequality and adding all the resulting inequalities together we find that A
I n-(q+l) < q-l(l-A-q) < 1/ q.
n=2 which implies that A
I n-(q+l)< 1 +O/q)={q+ l)/q,
Q.E.O.]
n=l
Comparing this with the result in 1°, where we ohtained an esti· mate for the case a1=a:;= ... =a l ",O, we reach the following conclusion:
LE MMA 3. Let l> 2 (OLd l;;;;A;;;; B;;;; c(l)A l-l. Then the sum of the numbers of solutions IZ i I; ; B (l~i;;;;l) of all equations of the form (9)
where lail;;;;A (l;;;;i~l), does not exceed c(l){AB)l-l.
§4 TWO MORE LEMMAS Before proceeding to prove the fundamental lemma, we have to derive two more lemmas of a special type. They are both very simple, in idea as well as in form, and yet their assimilation might cause you some difficulty because they are concerned with the enumeration of all possible combinations, whose construction is rather invol ved. The difficulty with such an abstract combinatorial problem is that it is hard to put it in mathematical symbols: one has to express more in
48
words than in signs. This is of course a difficulty of presentation, however, and not of the subject itself, I shall take pains to outline all questions that arise, and their solution, as concretely as possible. \\ e shall denote by .4 a finite complex (i. e., collection) of numbers, not all of which are necessarily distinct. [f the number a occurs A times in lhe complex A, we shall say that its multiplicity is A. Let a 1, a?, ... , aT be the distinct n umbers which appear in A, and let '\l,A" ..... Ar be their respective multiplicities (because the com-
ann
T
plex A contains all together .~\ numbers). I.et B be another com-
,=J
pie" of the same type, which consists of the distinct numbprs b 1 b',2, ... , b s with the respective mlJltiplicities fl1,fl", ... , fl s ' Let us investigate the equation (12)
,
x+y=c,
where c is a given number and x and y are unknowns. We are interested in such solutions lx, yl of this equation in which x is one of the numbers of the complex A (abbreviated x 10 A) and y is one of the numbers of the complex B (yEB). If the numbers x=a i and y=b k sati sfy equat ion (12), this yields \IL k sol lit ions of the required kind, because anyone of lhe >-; "specimens" of the number ai' which occur in the complex A, can be combined with an arbilrary one of the flk specimens of the number bk appearing in the complex B. Hut we have* Aiflk ~~(At+flV, Therefore the number of such solutions of quation (12), where x = ai' Y = bk' is not greater than ~~(,\} + flV, ) t follows that the number of all solutions xEA, yEll of equation (12) is nol more than the sum ~~(A7 + flV, J1l're the summation is over all pairs of indices Ii, kl for whi ch a i + bk = c. Our sum is enlarged if we over all i and fl~ over all k (because every bk can be comsum bined with at most one a i .) It finally follows, therefore, that the num-
'\t
ber of solutions x 10/1, y EB of equation (12) does not exceed the number r
1'( /2
S
~\2 .... fl..
2)• + " ~ iJ-, i=l' k=l'
"'''The geometric mean is not greater than the arithmetic
mean'~.
plest proof:
O;i,(\-/1k)2=A;+fl~-2\ILk'
and hence
2'\h~At+IL~.
Here is the sim-
49
On the other hanel, let us consider the equation (13)
x-y=O
and calculate the !lumber of its soilltions x fA, Y fA. Clearly every such solution is uf the form x=y=a i (] ~i
AI
tain sol nlinns, beeallst" tht" nwnbf'rs x and y can coillcirit", indepcnrlenLly of one unother. with anyone of the >-; specimens of the number a j appearing in 1. \ccordingl y the total numher of solutions x E I, y,ay
Y El of pqnation (l:n i" equal to
i ,\2.
j-[ ,
In exactly the ",amP
we finrl. of course. that the number of solutions x E 11,
" the same cqudlion is equal tu ::SIL~. k~
y f n of
[f \\c compare these results \\itl!
the ()nc f()unrl abovc, \\c reach the follo\\ing con( lu"ion:
LJ:: ~HI.\ 1.
j
he number of solutions of the equalion x
does not exceed hlllf quations
lite
-l)
sum of
C,
x E 'I. y E 8
Ihr:
numbers of '>u/ulwTls a/lite e-
'I, yEA
x-y=o,
xE
x-y=O,
XEB, yEn.
and
For the special c"!oe in \\hich the eumplexes ;I and 11 coincide we obtain the follo\\ing COltOLLARY.
The numberofsolulions ofllle equalion
x+y=c,
xEA, yE 1
rioes not exceed the number of solutions of the equation
x-y=o,
xEA,yEA.
[',ow let k and s be two arbitrary natural numbers. We' put /C.2 s =l, and investigate the equation
J ,et .4 1. A 2,
..• ,
A I be finite complexes of numbers. Suppose that
the complex Ai (I~i~l) consists of the distinct numbers a i l'a i2 , •..
50 with the respective multiplicities Ail' Ai2' ••.. We are interested in the number of solutions of the equation (14)
If we set %1 +%2+···
+%112
=%,
%(l/2)+1 +···+%1 =
Y
(l12 is of course an integer), then the given equation can be written in the form %+y=c,
and Lemma 4, which we have just proved, can be applied to it. We have only to find out to which complexes the numbers % and y belong. Since %i e:A i (l ~i ~ l), % can be an arbitrary number of the form Z1 +z2+ •• ,+ z l/2' where zi EAi (l~i~l/2). Similarly y can be an arbitrary number of the same form, where, however, zi EA(l/2)+i (1~i~lI2).
Hence, by Lemma 4, the number of solutions of equation (14) does not exceed half the sum of the numbers of solutions of the equation (J5)
%-y=O
under the following two hypotheses: 1)
% = Z 1 + Z 2 + ••• + Z l/ 2'
y=z; + Z~+ ••• +Z[;2'
where (16)
2)
%
and y have the same form, but
(17)
In both cases equation (15) may be rewritten in the form (18)
We conclude therefore that the ntunber of solutions of equation (14) does not exceed half the sum of the numbers of solutions of equation
51 (1S) under the hypotheses (16) and (17), i. e., it does not exceed half the sum of the numbers of solutions of the equations l/2
OSa)
!. (z .-z.')=O,
z,' €A i ,
112 !(z.-z.')=O, i=1' I
Zi EA(112)+i'
i=1
'
,
and (I8 L)
z[€A(l/2)+i
(l~i~l/2).
E'quation (18) has [/2 summands on the left-hand side, i. e., half as many as the original equation (14.). We set l/4 ~ (z. - z~) = x, i=1' •
l/2 ~ (z.-z·')=r, i=(l/4 )+ 1 ' ,
anrl therehy hring equation (IR) into the form x+r=O.
To Ihis \Ire can app Iy J ,t'mmu ~ anpw. It is evident that, just as we arrived at equation (18) from equation (14), we now get from equation (1 R) to the equation 1/4
~ (u.+ u.'-IL."-U .''') =0,
(19)
i=1'
,
,
,
where we have to consid.er the sum of the numbers of solutions of this equation under the follo¥oing (now four) hypotheses:
Since l =k.2 s • we can repeat this process s times. We evidently end up then with the equation k
(20)
!.lrP )+rF)+,.. +r~25'1) _rPS-l+ll_ ... -rFsl! =0, i=1
where we have to consider the sum of the numhers of solutions of this equation under 2 5 different hypotheses, viz.:
52
............... 2 S ) r~j)
e:A k2 s_ k +I'
···,rV) e:A k2 s.
If we put r(j)=rfil +r2(;) + ... +rk(j)
then equation (20) tales on the simple form
r( 1) + y(2) + ••• +y(2 S - 1 ) _yt2 S - 1 +l) _
(21)
••• _y(2 S
)
= O.
Here we are concerned .... ith the sum of the numbers of solutions of equation (21) under the following 2S hypotheses, which differ from one another in the value of the pHrameter II! (O~u ~2s -I): y(j) =Y1(il
... y iil + ••• +Yk(j),
where A 2S ) • Y1(jl e: /'wk+l' Y2(i) e: A wk+2' ••• , Yk (j) e: .4 tw+l)k (.J= 1') ,~, ••• ,
Thus we can express the 6nal result of our deduction in the form of the following proposition: LEMMA
5. If l=k·2'<, the number of solution!; of the equatLOn
(H)
does not exceed the sum of the numbers of solutwns of the equation Y( 1) +y(2) + ... + y(2 S
(21)
-1) _y(2$ -1+1)_ ••• _y(2 S
y(j) =Y1(j) + Y2(j) + .•• + Ylc(j), (j) A (j) A Y1' e:;,wlc+l' Y2 e: wk+2' ... , Yk e: (w+l lie (.)
A
)
=0,
}
(j=l,2, ... ,2 S
)
under the hypotheses u:=O, 1, ... , 2s-1. Notice the connection between Lemma 4 and Lemma 5 for
k=s=l. l=2. This winds up our preliminaries, and we are ready now to hegin the direct assault on the fundamental lemma.
53
§S PROOF
OF
TilE
FUNDAMENTAL
LE~BI'\.
\\(> arc going to prove the fundamental lemma by the method of induction on n. It is often the case in inductive proofs, that a strengtht"n ing of the propOS1 tion to be proved, considerably [aei Iitate~ its proof by the gi ven method (and sometimes is actuali y \\ hat 1Ilakes the proof feasibll" in the first place). The reason for this is easy to understand. In inductive proofs, the proposition is assumed to be correct for the number n - I, and is proved for the number 71. lIenel', the stronger the proposition, the more that is given to us by the casl' n - 1; of course, so much the more has to be proved for the Humber n, but in many problems the first consIderation turns out to be llIorc important than the sec one!. ·\nd S0 it is, in fael, in thf' prpsenl case. Of immediate interest for us is the nllmber of ::.olutions of the e'1uation :t1+x~+ ••• +'tA=m (l ~m~I\) lv.hpre, aecorclin~ to the \'ery ml"uning of the problem, O~xi~ml/n~.\'t/n). nUI.t" is tht" simplest special ca::.c of ann-th de~ree
pol ynomi al
[C,,) =ac)"'t n ~ (II x n . 1 + ... +a" .1 x + an' and it wi II bt" to ollr aclvantage to replace the given equation (I) by the more general equation (2~)
[(Xl) +!(X2) + ... + (b'-Ie) =m,
where the unknowns are subjected to the weaker conditions IXil ~ NIln (] ~i~k)' The proof of our proposition for equation (22) will give us more than we really need; bllt, us you will sec, it is just this strengthening of our proposition .... hich creates the possibility of an induction. And so, for m ~N. leI us rlt'llOlf' by rk(m) tht' number of solutions of eqllation (22) .... hieh satisfy the conditions Ixil ~/\ lin (1~i~k)' Of course \\e are still free to dispose arbitrarily of the coefficients of the polynomial {(Y;) in the interest of the induction to be performed (provided only that the imposed conditions are satisfied in the case f(x)=x n ), \\e are going to prove the following proposition:
Let the coefficients of the polynomial {(x) !wtis[y the inequalities
(23)
54 Then, (or a suitably chosen k=k{n), rk{m) < c(n) N(kln )-1
(1 ~m ~lV).
Since the inequalities (23) are obviously satisfied In the case f(x)=x n for c{n)=l, this theorem is indeed a sharpening of our fundamental lemma. Let us first consider the case n= 1, j{x)=aoX+a1' "e set k(l)=2, so that equation (22) acquires the form
aO{x1 +x2)=m-2a1' We are inlt'rcsted in solutions of this equation which satiRfy the requirements IXll ~N. IX21 ~N. Thus at most 2N t 1~3N values are possible for x l' But at most one X2 corresponds to every x 1, so that
which completes the proof of our proposition for n=l (k=2). Now let n> I, and suppose that our assertion has already bccn verified for the exponent n-l. Put k{n-I)=k' and choose
k=k{n)=2n.2[41og2 k
'1,
where the exponent means the greatest integer not exceeding 410g 2 k: In the sequel we shull set ['~log?k'J-l=s, for brevity, so that (24)
To estimate the number, rk{m), of solutions of equfltion (22), we first apply Lemma 4 to it, setting ~k
x =."1. {(x), .=1 The complex A (and the complex R which coincides \~jth it in this case) consists of all sums of the form ~k
i:-/{x i ),
where
IXil ~Nl/n
(I;;;i~ ',~k).
By the Corollary of Lemma 4, rk em) does not exceed the number of solutions of the equation x-y=O. where x EA. rEA, i. c.,
55 kl2
kl2
x=.I[(x), y=.I[(y),
,=1
,=1
In other words, rk(m) does not exceed the number of solutions of the equation (25) ",here IXil~N1In, Iytl ~N1/n (l~i~k/2). We now set xt-Yt=h t (l~i~k/2) and replace the system of unknowns IXi'y;l by the system {Yi' hJ: here we allow y i and hi (l ~ i ~ "/2) to assume all possible intewul values in the interval <_2N1/n,+2N1/n>, which can only increase the number of solutions of our equation. This means that every summand [(Xt) - [(Yt) in equation (25) is replaced by the expression n·l
[(y.+h.)-[(y.)", ~a I{y.+h.)n-v_y .... v} 'v=O v " , " n -1 n-v
= ~a
~(ntv)h~y~-v-'.
v=O v /=1
' •
If we change the variable t of summation by pllttin~
v+t=u, so that
n-v-t=n-u,
t=u-v,
we obtain
n-1 n [(y.+h.)-[(y.)=h. I a I (n-v)h~·v-1y~.u •• , • v=o v u=V+1 u-v. • n u-l =11.. I y,!-u I a (nu-v)h!,-v-l • u=I' v=o v -v , n ~ a. y~-u=h.¢.(y.), 'u=l '. u l. r.' r.
=h. where
56 is a polynomial of degree n - 1 with coefficients u-l a. = I a (n-v)h~-v-l (] ~i~k/2) '.U v~O v u-v , which depend on the numbers hi" Thus, In the ne .... variables IYi' form
hJ,
equation (25) assumes the
(26) In this equation the numbers hi and Yi may take on arbitrary integral values in the intcrval <_2IV 1 / n ,+'llV 1 / n >, \\hclc we must Lear in mind that the coefficients of the polynomial::; 1>/y) (of degree n-l) dppend on the numLers h. Marl well that we have proved the following so far: The number rk(m) which we are estimating, does not exceed the sum of the nlLmbers of solutions in integers Yi' Iyil ~2fvl/n (l~i~k/2), of all the equations (26) which can be obtained from all possible values of tlte numbers hi' Ihil ~zVI/n (j ~i~k/2).
§6 CONTINUATION
~ie
are now going to examine one of the equations (26), i. e., we shall regard the numbers hi (1 ~ i ~k/2) for a whil (" as fixed. Let us apply Lemma 5 to this equation; the nllmbers hi¢i(Yi) play the role of the unknowns xi' the number ~2k=2n.2s plays the role of the number l, and we set 2n=ko for brevity. H.ccall once more that the numbers hi appear in equation (26) not only explicitly but also through the coefficients of the polynomials ¢/Y). The complex Ai to which the numbers Xi =h/Pi(Yi) must belong consists, in the present case, of all numbers of the form hi¢i(Y t )' where the numbers hi have given, fixed values and the numbers Yi run throngh the interval <_WI/n, +2Nl/n>. According to Lemma 5, the number of solutions of equation (26) satisfying the requirements just described, ooes not excced the sum of the numLers of solution::; of the equation (21)
yO) +y(2) + ... +y(2 S -1) _y(2 s·l +1) _
... _y(2 S )=0
under the following 2 S hypotheses \\hich correspond to thc values
57 of the parameter u =0,1, ... ,2 8 -1: (jl, ),(il="J l(j}-ty "'?
1
•••
Iy(j)} Jko'
,(l
Ar (1 ~r~2s) is the complex of nlllllbf'rs of lhe Conn hr¢/yrh,ilit prescribed hr and arlJitrary Y r' \),,1 $ 21\ lin. For the CH!;C II =0 (\\hich Wt' choose mere}) H!:- an example), equation (21) in expanded frmll looh as follows:
where, rernembl'f,
ly~1)-t y}1l
t ...
+Yi~ll
+ lYl(2)fY2(2l+ ... +y~;ll
+. . . . . . . . . . + ! Yl
(2 s-l)
(28-1)
+Y2
(2 5 ,1)1
+ ... f.jko
(8-1 l 8-1) +1 +Y2 2 +l + ... +yi; f-l J
( s-I)
- !Yl
2
nr, rc,uTanging the summands. lyP)+yPl+ ... lyps-ll_Y1(2S-1jl)_"'_Y1(25ll + IY2(1)+Y2(2)+ ...
+yps-ll_y~ps-l+1
)-"'-Y2(2")1
+ .... +
!r,,(l) -t Yk(2 )+ ... +y,,(2 S- I ) _y,,(2 o 0 . 0 0
S -
1+1)
_ ...
_y,,(2 S ) 0
J-o;
everyone of the numbers yY) is a numher of the fonn hi~Ji(vF)), where
1
vi(j) 1 ;;:2\ 1/". llen('e the lagt equation ('an be rewritll'n in the
fonn
h 11
Uy pulling, for brevity,
))
I + ... + hk 0 l
58
this equation can be written quite shortly as follo'~s: (27)
All together we have 2 S equations of this sort, and their totality can be written down in the compact form ko
'£ h k +,z ko+'=O i=1 W 0 'w , For the present, however, we shall confine our investigation to equation (27), which may of course be regarded as typical. To estimute the number of solutions of this equation which interest us, we must first see within what limits the quantity ¢;(vi(j» can vary. To this end we recall that (p.55)
where u-l
= v=O '£ a (n-v)hlf- v - 1 t.u v u-v ,
a,
1
lIence it follows from our hypotheses lavl
i. e., in view of (28)
u~n,
lai,,,1 < c(n)N(u-l)fn.
On the other hand, because of Iv~j}1 ~21V1In, we have Iv~j}ln-u~c(n) • N(n-u)1 n and consequently Ia i • u 1'lv?)1 n-u~ c(n) N( u-l )/n N(n-u)1 n=c(n)N(n-l )/n. The same estimate (with another c(n» holds for the whole ¢i(v;(j}), since the number of terms of this polynomial is equal to n. Accordingly
59 But every zi is the sum of 2 5 =c(n) summands of the form ±¢/vP'), and therefore
( .... ith another c(n). naturally). This means that in equation (27) every zi can u'jsume only the values lying in the interval <-c(n)l\{n-ll/n, I
c(n).\'(n-I)/n>.
Let m b(' nne of these numbers. The equation Z i =m can Le salin general not only in one hut in several way!'!, hecause the definition of the number zi (p.58) is such that one and the SdIne value of zi can very ..... ell result from different choices of the nllmhers vii> (1 ~j~2$). \\e now have to estimate the number of solutions of the relation zi=m, i. e., of the equation i~lied
"'.( v.(l » + •.• + ¢.( v( 2 5 -I» _'"
(29)
'PI.
,
r.
.(v.(2 5-1+ 1)
'PI.
I.
r.
_ ... _¢( v(2 5 » t
t
=
m.
For this purpose .... e shall finally have to apply the long-promised induction. \\e proceeo as follows. First we rewrite equation (29) in the form
=
m-'"'-Pt,.(v.r.( k '+1)
_ ••• +"'.( v.{, 2 'f.lt
5-1
This is possihle because f,)f Ie' = already k(J) =2) we have
($)
+1) + ••. + "'.( '1-'1. v.I. 2
ken -1) > I
2 s - 1 -_2[t !og2k']-2 >,1:"
) •
(und we have seen that
•
(In detail: k'"?,2, log2!c'"?,1, :nog'2k'"?,3, [1Iog 2 k']-2>11og 2 k'-3"?, /") k ' 2s_I_2l41og2k']-2 I Og2, "?, I: • If we denote the right-hand side of the last equation by m', ,,'e
get (30)
I,et
liS
choose some particulur values for the numbers
t,P)
(k'+ 1 ~j~2s)
(in the interval <_'lNl/n,+2';I/n>. naturally); then ",'also acquires
60 a definite value. To equation (30) we now apply the theorem to he proved, since ¢j(Y) is a polynomial of degree n-1. \\e have to verify that all the necessary hypotheses are fulfilled. \\e have n
¢.(y)= ,
la. yn-u, u=l I. u.
\\ here, aecordi n/1; to (28), u-1
(31)
n-1
u-1
la.'0 u I < c(n)l't-n-=dn)(I\-n-)Ii"=T,
and, as is easily seen, n-1
1m ' 1< c(n)!''' --;r(because m and all chj(yF» satisfy this inequality). In virtue of the last inequality, the role of lv' can be assumed by the number c(n) N(n-1 lin; then the conditions (30, whi cli the coefficients of the polynomial ¢/y) satisfy, are precisely the conditions (2.1) wi tb n rep lacE"d by n - 1. Thus all the hypotheses are indeed fulfilled, and we can assert that the number of solutions of equation (30). for ¥.-hich lui{j)1 ~Z,\'1/n=2(N(n-1lIn)1/(n-l~ does not exceed the number
l!.:!
(32)
L _1
c(n.}{ JV n )n -1
= c(n} tI,'
I.. '.n+l
n
•
.
This estimate IS obtained for the fixed values u.u<;'1l, .•• ,v.(2 , S Clearly we have at most
).
(33)
such systems of values. The total number of solutions of the required type, of equation (29), therefore does not exceed the product of the right-hand sides of (32) ,md (33), i. e., it is at most
We now return to equation (27). \\!e say, before (p.59) that every can assume only the values lying in the interval <-c(n),\ (n-ll/n, + c(n)N(n-l)/n>. :-.iO\\ we sec that the "multiplicity" of each of these zi
values (i. e., the number of ways of choosing the yEi> so as to snt-
61 i 'if ~
th e equat ion) doc", not
t'X
cecil the nulTtber (:3 I).
This result makes II pos!--ihlc to reduep the \\hole rrnlolem to an c:,:timalion of Ihe nUI1lLf']"-; (.f :"olutions nf linear ("IUalioll,... I,'or .:It the t!nd of ~S
\\1'
r .. dllcp,1 Ihe c~timution of rk(m) 10 Ihc! c:sLlllIuLinn
of the number:s of "oluti()n,.. of equations of the fOl'm (2()). 1~lIt l.~
prc)\(·.(
ae; \\('
an appllulllfllI ,,/ 1.l'mma 'i. Ihe number of "'Olllliulls of
lx,
e,!uall'"1 (;2(,), for whl('h
lOr
tire 1IIlIIlbc-r ...
:;:2,\
till' "'"111
I n. l'i at IIl(lsl (''1l1nl In
0)[
s"llIllon'" (If ~s '!«llall()JI" of Iype (27), I. c., .. lrc<.1d}
line',l1 (''!"<.1linn",. In thiS COIIII('('lion w(' "blainpr/ lillllt-; \\lthin \\hich thc ullkllu\\n::. zi ,Ire allo\\c.\ to priCe'
hllv(' had
Wf>
that Ih.:
10
\.JI").
\
certain ne\\ dIfficulty (the
ra~ f()r thl' Iran"ltinn to lin'! ...' P.'JlHltlOlI")
IInknn\\lls z. hU\e' Lo Le con-;jticre'd
IIC\\
\\111.
I'"
r('r[aill I11UItI-
rlil'itil's (fnr \\hie'h \\\: havc .. Isu determin.'.( limits). l'illall}
\\t'
mu,..tllol fnrgl'llhaL all till',,!, r,ll(ul,lllLlns nrc'
\Jlldcl' Illc a:s,..nmption Lhllt tlr.~ IllJlllbcrs Thl'rdorc
\\l'
IIHHle
hi arc cho:scn ,lOti fixed.
,..till have' In UlI.!llpl}' Lhp rf>c;l\lt (lbtollllt,c1. b} Ihe
1111111-
uer nf all 'illl h I'o". ... ibl.· dllll ce,... The' final rt'!-olllt of Ihlt' ,.,e( lIon, "hich \\e ha\ c 1(. kpep in rllind. re
of solutio/l" ill
integers zi'
Izjl ~c(n),\(n-I )/11.
pitelLtcs Ai~c(n),\(2'-"'1 ):", of l''1l/al/fl/ls uf ko ~h,
(35)
,~l
l/'here
I/,
11 h."(J -{
,zk
I
() I ,
W
runs th rough the l'fIlues 0,
fAith lIIulti-
,hI' fom!
~(),
J , ... ,
2 s -1. and (11(' /1/11" b,'r~ h,
(l ~r~~'ko) a'SIIIfl(,. ind('Jl<'ndcnli) of OlL(' IlIIOLlier, all Ifl(cg('rs m the inten'al <-21\ l.'n. ' ~\ lin>. \nJ ~o \\ e se(' that
\\C
Ira\ e nuw obt
in whose formulalion the gi\'Pll pol} nomiul fix) doc,," which lends thib eblimaLc a
Vf'I'}'
r" (Ill),
"pp"
gcneral LhLU'd,c[e'I'.
CO'lCI I\U\\
11"[
lISIO~
that \\P have' reduccd the problem to an (·,.,limatlol1 of the
nwnber qf solutions of linear e,/uaLioll's \\hiLh are independent uf the special fonn of the pol}nomial fix), .... e quickly re'Hlh
0111'
goal "ith
r
the aid of ,COlma 3. Denote
Ly
l\. any particular C'omhin.llilln of Ihe numb!'r:; h,.
62
Ihil ~r2Nl/n (l~i~k/2), and by Uw(K) the number of solutions of eq'lation (35) for this fixed combination K and for a certain prescribed w, where we are concerned with those solutions Zt which satisfy the inequalities Izil ~c(n)N(n-l)/n, with multiplicities \:;:,; c(n)N(2 s -n+l)/n. Then, according to the final result in the preceding section, T-l rk(m)s ~I I U (J...) I, - K w=ow where the summation over K extends over all admissible combinations K of the numbers hi' This can be written T-l rk(m) ~ w::o I l;L' tK)\.
r
It is immediately evident, ho .... ('\"~r, that for different u the sums ~ Uw("') do not differ from one another at all (because for different K
w the equations (35) do not dill"er from one another in any respect).
\\ e can therefore write rk(m)s2S~U(1(K)=c(n) ~ Uo(I\). -
K
K
Here Uo(K) is the number of solutions of the equation
(36)
h1 Z1
+h 2 z 2 + .• , + hkozko=O
for the given combindtion K of the numbers hi' Ihil ~2Nl In (l~i~/2), .... here Izil ~c(n)N(n-l)/n and the zi have multiplicities \~c(n). N(2 s -n+Uln. Let us denote by U't(K) the numher of solutions of the same equation under the assumption that all zi are simple. Then clearly
or, recalling that ko=2n,
Vo (K) ~ c(n) N2(2 s -n+!) U~(I\), amI hence (37)
Now let us note the following.
~;very
K represents a certain admis-
63 sible combination of the values of all hi (] '$.i~J[/2); the number U~(K), however, is completely determined by the values of the fin,;t ko =2n of these values (l~i~2n), because they alone appear in equation (36). Of course \\'hen we choose a certain fixed combination K, we thereby also uniquely define a certain combination h.' of the values hi> h2' ••• , h 2n • nut if, conversely, a certain combination 1\.' of the numbers hi> h'}., ... , 1t2n is selected, then' corresponds to it not the single combination 1\, but ruther as m.my as there arc ways of choosing the remaininp; "supplements" hi (21£<1 ~k/2). Since every hi must belong to the interval <_2N lln , I 2Nlln>, it is eviden t that to a combinat ion 1\ ' therc correspond at most c(1£) (.\, 1/ n )(k/2 )-2n =- c(n) .V(k /2" )-2
combinations K. Ilence ~ U~(I\)~c(Tl)N(k/2n)-2 ~
1.,
where U~(I\. ") is the number of solutions ill Integers zi' Izil'$.d1£)' N(n-llln n~i~2n) of equation (36) for the given combination 1\ ' of the numbers hi' Ihil~2 \lln (l~i~2n), and the summation is to bl:' extended over all such combinations. From (:37) \~e therefore obtain* (38) r. (m)~c(n)N2(2s-n+l)i\(k/2n)-2 ~ li8'
-
... '
K'
0
Finally, ~,U*(l\.') is immediately estimated with the help of Lemma 3, wh~re we have to put l=2n, A=2N I/ ", R=c(n)/\,(n-ll/n; you can easily verify that all the hypotheses of Lemma 3 urI:' satisfied. On applying this lemma we find that
I U*(K')s:c(n)(AB)2n-1 =c(1£)N2n-l. K'
0
-
At last, inequality (38) yields r/m) ~ c(1£)N2(2 s +l-n). y2n-l "" c(n)N2' 2 s+l_l =
c(1£)/\'~ -
\
which completes the proof of the fundamental IClllllla and therchy also of lIilbert's theorem. This prouf, so exquisitely elemeDtary, \\ill unduubteJly seem very complicated to you. But it will tab> you only two Lo three ·Recall that k=2n·2 s + 1 •
weeks' work with peneU and paper to UDderstand and digest it completely. It is by conquering dUficulties of just this sort, that the mathematician grows and develops.
* '*
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