On Applications of Van Der Waerden's Theorem
Author(s): John R. Rabung and Van Der Waerden
Source: Mathematics Magazine, Vol. 48, No. 3 (May, 1975), pp. 142-148
Published by: Mathematical Association of America
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142
MATHEMATICS MAGAZINE
[May-June
Supposethetimetofailureofeachofthecomponents
isexponentially
distributed
withmean1/1and supposethesystem
forexactlyT hourseachday.Now,
operates
whoselifetime
a component
followstheexponential
distribution
showsno aging,i.e.,
theprobability
thatthecomponent
survives
dayn + 1 giventhatitsurvived
dayn is
thata newcomponent
thesameas theprobability
will surviveday one. Hencethe
stochastic
analysisofthis(activeredundant)
is equivalentto the analysisin
system
Section2 providing
we identify
(i) thenumberof ringsr withthenumberof components,
and
of failureq on a singletoss of a ringwiththe
(ii) the(constant)probability
e-AT thata component
willsurviveday one (= T hours)
(constant)probability
The playin theringtossinggamecan thenbe identified
withthecomparison
builtfromtwodifferent
betweencompeting
systems
typesof components.
ON APPLICATIONS OF VAN DER WAERDEN'S THEOREM
College
JOHN R. RABUNG, Randolph-Macon
1. Equivalentversionsof thetheorem.In [1] B. L. van der Waerdenrelateshow
Artin,Schreier,and he were able to findthe proofof the followingconjectureof
Baudet:
(A) If theset ofpositiveintegersis partitionedin any way intotwoclasses, then
for any positiveintegerI at least one class containsa set of I consecutivemembers
ofan arithmeticprogression.(Henceforthwe shall use thephrase "arithmeticprogressionof lengthl" to mean a set of I consecutivemembersof an arithmetic
progression.)
Aside fromthe ingenuityof the proof whichfinallyarose, one of the most intriguingaspectsof the paper is the mannerin whichthesemenwere able to manipulate Baudet's conjectureinto more manageable,yet equivalentforms. The first
suchmanipulativestepwas to considerthefollowingstatementsuggestedby Schreier:
(B) For any positiveintegerI thereexistsa positiveintegerN(l) such thatif the
set {1,2, ..,N(l)} is partitionedinto two classes, thenat least one class contains
an arithmeticprogressionof length1.
This so-called"finiteversion" of (A) clearlyimplies(A) and the converseimplication is shownin [1] usinga Cantordiagonal approach.
From hereit was an easy stepforArtinto show that(B) is equivalentto:
(C) For any positiveintegersk and I thereexistsa positiveintegerN(k, 1) such
that if theset {1, 2, ..*, N(k, l)} is partitionedintok classes, some class containsan
arithmeticprogressionof lengthl.
This is the statementwhichvan der Waerdenproved.(See [1] or [2].)
Since theappearanceof theproofseveralapplicationsof thetheoremhave been
published.(We shall discussA. Brauer's applicationto powerresiduesin another
sectionof thispaper.) However,it has become apparentthatthe real potentialfor
1975]
ON APPLICATIONS OF VAN DER WAERDEN'S THEOREM
143
applicationof (C) maylie in thesize of thenumbersN(k, 1). In his proofof (C) van
der WaerdenconstructsnumbersN(k, 1) whichsuffice,
but it is thoughtthatthese
numbersareterrifically
loose. For example,van der Waerdenconstructs
constructed
N(2, 3) = 67 whereasany N(2,3) ? 9 will work.And just a glance at his general
of thesenumberssuggeststhat theirgrowthrate is muchgreaterthan
construction
it need be. As Erd6s pointsout in [3], tightening
of the numbersN(k, 1) may lead
to settlingthe questionof the existenceof arbitrarily
longstringsof primenumbers
whichare consecutivemembersof some arithmeticprogression.
of thenumbersN(k, 1) willnot be achievedwithoutan essenBut the refinement
tiallynew proofof (C). Because none have yet been found,one is led back to the
reasoningof Artin,Schreier,and van der Waerdenthat perhaps anotherversion
of the statementwould be more manageable.Severalequivalentformsof (C) have
appearedsincethne
proofwas published.In thissectionwepresentsomeotherequivalent versionsof (C). These versionsseem more explicitlyrelated to the problem
of primesin arithmetic
progressionthandoes the statementof (C). We beginwith:
(D) Let S = {ai}1=, be any strictlyincreasingsequence of positiveintegers.
If thereexistsa positiveintegerM such thatai+1 - ai < M,for i = 1,2,*., then
thereexist among the membersof S arithmeticprogressionsof arbitrarylength.
This is quicklyseen to be a consequenceof (C), forconsiderthe followingpartition of the set of positiveintegersinto M classes:
Ko = {aj: al E S} = S
K1 = {ai + 1: aieS}
Ko'
K2 = {al +2: aieS}
(Ko UK1)'
Kj = {ai +j:
(Uh-1 Kh
a1-S}n
KM-1 ={ai + (M-1):
aeS}
n(UM_1
Kh)'.
It is clear fromthe natureof S and the construction
of the classes, K., thatthisis
indeeda partitionof the set of positiveintegersinto M classes. Now by (C) there
must be an arithmeticprogressionof length1 (arbitrary)in some class, say K,.
Suppose this arithmeticprogression is {b, b + d, b + 2d, *. , b + (1- 1)d} . Since
b E Kn, we have b = a + n for some a E S. Similarlyeach memberof this progressionis n greaterthansomememberof S. Thus,{a, a + d,a + 2d, ., a + (1- 1)d}
c S, and theresultis obtained.
Now (D) immediatelyyields:
(E) If thesetofpositiveintegersis partitionedintotwoclasses, thenat least one
of thefollowing holds:
(1) One class containsarbitrarilylong stringsof consecutiveintegers.
(2) Both classes contain arithmeticprogressionsof arbitrary length.
This is clear since if (1) does not occur,then(D) applies in both classes. And
144
MATHEMATICS MAGAZINE
[May-June
since(E) readilyimpliesthe Baudet conjecture(A), we see that(D) and (E) are each
equivalentto van der Waerden's theorem(C).
Now (D) and (E) both yield "finiteversions":
(D') For any M and 1 thereis a positiveintegerNd(M,1) such thatany strictly
increasingfinitesequence {ai}l71 ofpositiveintegerswithdifferences
boundedby
>
M (i.e., ai+1 a,< M) and witham- a,
Nd(M,1) will contain an arithmetic
progressionof length1.
(E') For any M and I thereis a positiveintegerNe(M, 1) such thatwheneverthe
set {1, 2, ,Ne(M, l)} is partitionedinto twoclasses at least one of thefollowing
holds:
(1) One class contains M consecutivenumbers.
(2) Both classes contain arithmeticprogressionsof length1.
From eitherof the statements(D') or (E') one sees the connectionbetween
van derWaerden'stheoremand theproblemofprimesformingconsecutivemembers
of an arithmetic
progression.If one can sharpenthe numberNd(M,1) enoughand
observea relationshipbetweenthisnumberand the rate of growthof gaps between
consecutiveprimenumbers,one maybe able to settlethequestion.
We have not beenable to generallysharpenan estimateforNd(M,1),but we have
foundbestpossiblevaluesofNd(M, 1)forsomevaluesofM and 1.These are presented
in section 3 of this paper.
2. An applicationof Witt's generalizationof van der Waerden'stheorem.Very
shortlyaftervan der Waerdenpublishedhis proofin [2], A. Brauer [4] showed,
among otherthings,thatforany positiveintegersk and 1 thereis a numberZ(k, 1)
such that for any prime p > Z with p 1_ (modk) the reduced residue system
p -1} modulo p contains 1 consecutivenumbers,each of whichis a kth
{1,2,,
powerresiduemodulop. Severalauthorssince have found uniformupper bounds
on thisstringof consecutivekthpowerresiduesforfixedk and 1. (See forexample,
[5]-[10].)
J. H. Jordan[11], withoutthe assurance of a generaltheoremlike Brauer's,
steppedintothedomainof Gaussian integersZ[i] and proceededto findseveraluniformupperboundsforwhathe called "consecutive"Gaussianintegerswhichare all
kthpowerresiduesof a primeof sufficiently
largenorm.The questionarises,then,as
to whetherthereis a theoremlike Brauer'sforthe Gaussian domain.In thissection
we show thatthereis such a theorem.
It is ErnstWitt'sgeneralization
[12] ofthevan derWaerdentheoremwhichallows
one to use an approach analogous to Brauer'sin not only the Gaussian integers,
but in some otherdomainsas well. Witt'stheoremmay be statedas follows:
Let F = {Yi' Y2' -) y be a fixed set of Gaussian integers. For any positive
integerk thereis a positiveintegerN(k, I) such that if the set
A
j=l
ajyj: aj eZ, aj _ 0,
j=1
aj1= N(k,l)}
is partitionedintok classes, some class will containa homotheticimage, F', of F.
1975]
ON APPLICATIONS OF VAN DER WAERDEN S rHEOREM
145
(Here we shall say that F' is homotheticto F if F' = Ar+ a = {Ay+ a: yE F}
whereA is a positiveintegerand a is an arbitraryGaussian integer.)Now if in the
above statementyj E r is such that yjI jyi|, 1 < i < 1, thenwe see
|j=1 aYj|
<
I
j=l
(|aj|Iv.I)< LvIj=lzjaj
=
LYjIN(k,l)
sinceaj > 0. Thus,we mayalso say thatifthesetofall Gaussianintegerswithnorm
not greaterthan (I y.,jN(k,1))2 is partitionedinto k classes, then some class will
containa homotheticimageof F. Let N(k, F) = (max{jyjl: yjE F})N(k, 1). Now we
can prove:
THEOREM 1. Givenany finiite
set of Gaussian integers,say F = {Y1iY2' ..1Y
large Gaussian prime, Xt,thereis a set P = {P19P2, "'Pi}
and any sufficiently
of quadratic residuesmodulo ir such thatP is a translationof F.
Proof. Let D(F) = maxtjIyjyi, yj e-F be called the diameterof F. By
choosingour prime7rwithlarge enoughnormwe can imbed any set of Gaussian
integershavingfinitediameterin a reducedresidue systemmodulo 7r. (See Hardman and Jordan[13].) Hence, we see thatforir withsufficiently
largenormthereis
eithera translatedimage of F close to the originconsistingentirelyof quadratic
residuesor thereis a quadraticnonresidue,v, modulo7t suchthat1 _ j vj < D(F) + 1.
Let R representany finitearrayof Gaussian integerswhichis largeenoughto contain some translationof F and vF = {vyj: yjE r} .
Now consider it to be a Gaussian prime such that jir j > 2N(2,R) + 1, thus
assuringthat a reducedresiduesystem(of the Hardman-Jordantype)modulo 7r,
imageofR in one oftheclasses.
whenbrokenintotwoclasseswillcontaina homothetic
In particular,eitherthe class of quadraticresiduesor the class of quadraticnonresidueswillcontainsuch an image,say R' = AR+ a forsome nonnegativeinteger
A and a a Gaussian integer.Now if A and theelementsof R' have thesame quadratic
naturemodulo 7r,then multiplication
by A- 1 moduloiryieldsR" = R + A- 1 consistingentirelyof quadraticresiduesmodulo it.
If A and the elementsof R' are of differing
quadraticnature modulo 7r,then
=
R" R + A-'a consistsof quadraticnonresidues.But now R + A la containsas
a subset some translationof vF, say vF + c'. Since v is a quadratic nonresidue,
r + v- la' is made up of quadraticresiduesmodulo 7t, and we are done.
FollowingBrauer's argumentin similarfashion,one can establish
THEOREM 2. Givenanyfiniteset F of Gaussian integersand any positiveinteger
k, thereexistsa translationoJF consistingentirelyof kthpower residues modulo
any sufficiently
large Gaussian prime ir withN(r) =1 (modk).
In theinterest
of movingtowardapplicationsof Witt's Theoremsimilarto those
mentionedin the firstsectionof thispaper, we note thatthis theoremhas the followingequivalentversionsanalogous to (D) and (E) of the precedingsection:
(A) If X = tail}' is any sequenceof Gaussian integersfor whichthereexists
a finiteset H = t{q1,q2,
qM} of Gaussian integerssuch that
146
[May-June
MATHEMATICS MAGAZINE
Z[i] c ? u (? + 'l) u (? +
u(
l2) U
+
qM)
thenfor any F = {Y1,Y2-') ,Y,} C: Z[i, ? contains a homotheticimage of F.
(D) oftheprecedingsection
is provedin thesamewayas statement
This statement
was proved.And from(A) we get:
(B) For any partitionof theset of Gaussian integersintotwoclasses one of the
followingmust occur:
(1) One class containsa translationof any finiteset of Gaussian integers.
imageofanyfinitesetofGaussian integers.
(2) Bothclasses containa homothetic
To establish(B) one needsonlyto observethatin sucha partitionifwe vieweach
eithertheconditionof(A) willapplyto both
class as a sequenceof Gaussianintegers,
sequences(and, hence,(2) holds) or thatconditionwill fail to hold forone of the
sequences.In thelattercase theclass K wheretheconditionfailswillhave arbitrarily
large"holes" in it in thesensethatforanypositiveintegerM we could find4 E Z[i]
such that
{ceZ[i]:
j4-4j ? MI nK
I
=
0.
M} in the statementof (A).)
(For, if not, we could use H = eCE Z[i]: I |
So (B), like (E) of theprecedingsection,bringsforththe questionof "patterns"
largegaps in the
of primeGaussian integers.Of course,just as thereare arbitrarily
large holes in the set of Gaussian
set of rationalprimes,so thereare arbitrarily
primes.But stillone is led to studya "finiteversion" of (B) as weighedagainstthe
growthrate of holes in the set of Gaussian primes.Here, as in the rationalcase,
such studieshave so faryieldedlittlefruitbecause of the unwieldysize of the constantsinvolvedin finiteversionsof Witt's Theorem.This is not unexpectedsince
Witt'sproofis essentiallythesame as thatof van der Waerdenin therationalcase.
results.Let {ai},' 1 be a strictly
increasingsequenceof positive
3. Somenumerical
integerssuch that for some fixedpositiveintegerM we have ai - ai < M for
i = 1,2, ..., rm-1. From statement(D') of the firstsectionof thispaper we know
of the existenceof a numberN(M,l) such thatif am -a _ N(M, 1), then among
the membersof the givensequencethereis an arithmeticprogressionof length1.
We directour attentionto the numberN(M, 1). Clearly,once such a numberis
found,anylargernumberwouldservethesamepurpose.Let N*(M, I) = min{N(M,1)}.
Underthisdefinition,
displayinga value of N*(M, 1) forsome M and I impliesthe
existenceof a sequence {ai}l=1 with differencesbetween successive members
bounded by M such that am- a, = N*(M, 1)- 1, and such that the sequence
progressionof length1. In presentingour numericalresults
containsno arithmetic
we shall also presentsuch sequenceswhichshow our constantsto be correct.Acassociatedwiththeoriginalsequence;
tuallywe shallgivethesequenceof differences
that is, if {ai}l= 1 is a sequenceto be displayed,we shallinsteaddisplaythesequence
m
m-1 . We shall also impose on our
{di}m'J=wheredi = ai+1 - ai for i = 1,2,
sequencesthe conditionthat di + di+1 > M for i = 1,2, , m-2. One easily sees
that this conditionin no way alters the generalityin computationsof N*(M, I).
1975]
ON APPLICATIONS OF VAN DER WAERDEN'S THEOREM
147
To give an easy exampleof how the computationswere made, let us consider
thecalculationof N*(2, 3). That is, we considerall sequencesof differences
di with
each di = 1 or 2 and withdi + di+1 > 2. Suppose d1 = 1. Then d2 = 2 and d3 = 1
are alike which,
or 2. If, however,d3 = 2 = d2, thentwo consecutivedifferences
of course, means threemembersof the originalsequence are in arithmeticprogression.So we considerthe case when d3 = 1. This means d4 = 2, and again
progressionof length1 = 3 in the
since d4 + d3 = d2 + d1, we have an arithmetic
originalsequence.This exhaustsall cases withd1 = 1. Similarargumentshowsthat
{2,1,2} before exhaustingall
with d1 = 2 one gets the sequence of differences
originalsequencemightbe {1, 3,4,6}. This is, in one
possibilities.A corresponding
sense, the longest such sequence with no threetermsin arithmeticprogression.
Since here, in the notation of the preceding paragraph,am-a1 = 5, we get
N*(2, 3) = 6.
The followingtable displays some values of N*(M, 1) which we have found
usingessentiallythe above techniqueand the CDC 3800 computerat the Research
ComputationCenter,Naval ResearchLaboratory,Washington,D.C.
N*(M, 1)
N*(2, 3) =
N*(3, 3) =
N*(4, 3) =
N*(5, 3) =
N*(6,3) =
N*(2,4) =
N*(3,4) =
N*(2, 5)-=
N*(2,6) =
{di}72j' of maximal length
Sequences of differences
6 1{2, 1,2}
18 {3,2,3,1,3,2, 3}
27 {1, 4, 3,4, 2,4, 3,4, 1}
64 {5, 4, 5, 3,5,4, 5, 1,5,4,5, 3,5,4,5}
102 {5, 6,4, 6,2, 6,4, 5,6,5, 3,5,6, 5,4, 6,2, 6,4, 6, 5}
15 {2, 2, 1,2, 2, 1,2,2}
57i {3, 3,2,2, 3, 3, 1,3, 3,2, 3,3, 1,3,3,2, 3, 3, 1,3, 1,3,2}
2,2,1,2,1}
2,25,1,2,
291 {1, 2, 1,2,2,25,1,2,
57 {2,2, 1,2,1 2,2,2,1,2,2,2,2,512,1,2,1,2,1,2,2,2,2, 1,2,2,2,1,
2, 1,2,2}
We also foundthepartialresultsN* (4,4) _ 160 and N* (2, 7) _ 193.
References
wurde,Abh.
von Baudetgefunden
1. B. L. van derWaerden,Wie derBeweisderVermutung
28 (1965)6-15.
Math.Sem.Univ.Hamburg,
NieuwArch.Wisk.,15(1927)212-216.
Vermutung,
BeweiseinerBaudet'schen
2.
theory,
Lectures
on Modern
3. P. Erdos,Somerecent
innumber
problems
advancesandcurrent
vol.3, T. L. Saaty(editor),Wiley,NewYork,1965,pp. 196-244.
Mathematics,
4. A. Brauer,UberSequenzenvonPotenzresten,
S.-B.Preuss.Akad.Wiss.,(1928)9-16.
Proc.Amer.Math.Soc., 16 (1965) 330-332.
5. M. Dunton,Boundsforpairsofcubicresidues,
Machineproofof a theorem
on cubic
6. D. Lehmer,E. Lehmer,W. Mills,and J. Selfridge,
residues,Math.Comp.,16 (1962)407415.
Proc.Amer.Math.Soc., 13 (1962)102-106.
7. D. Lehmerand E. Lehmer,On runsofresidues,
Canad. J. Math.,
powerresidues,
andW. Mills,Pairsofconsecutive
E. Lehmer,
8. D. Lehmer,
15 (1963) 172-177.
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[May-June
MATHEMATICS MAGAZINE
quarticresiduesmodulo
On theboundfora pairofconsecutive
9. W. Millsand R. Bierstedt,
a primep,Proc.Amer.Math.Soc., 14 (1963)628-632.
Math.Comp.,24
10. J. Rabungand J. Jordan,Consecutivepowerresiduesor nonresidues,
(1970) 737-740.
intheGaussianintegers.
residuesornonresidues
Consecutive
11. J.Jordan,
Math.Nachr.,6 (1951)261-262.
SatzderElementargeometrie,
12. E. Witt,Ein kombinatorischer
A minimum
problemconnectedwithcompleteresiduesystems
13. N. Hardmanand J.Jordan,
74 (1967)559-561.
Amer.Math.Monthly,
in theGaussianintegers,
ELEMENTARY EVALUATION OF 4(2n)
of Illinois
forAdvancedStudyand University
BRUCE C. BERNDT, The Institute
1. Introduction.Let Bj denote the jth Bernoulli number(definedbelow in
Section2). Euler's formula
(1.1)
((2n)
k-2n
k
-
I
)(2ic)
2(2n)! B2n,
(n > 1)
is one of the mostbeautifulresultsof elementaryanalysis.Perhapsthe threemost
common methods of proving(1.1) are by the use of the Fourier seriesfor the
Bernoullipolynomials[4, p. 524], bytheuse ofthecalculusofresiduesin conjunction
withthe Laurentexpansionof cotx (givenbelow) in termsof Bernoulli numbers
[10, pp. 141-143], and by the method of Euler, describedin [1], for example.
T. M. Apostol [1] recentlygave a proofof (1.1) thatuses knowledgeof symmetric
functionsand one of Newton'sformulas.The idea forApostol'sproofcan be found
in the Yagloms' book [14, pp. 131-133], althoughthey only establish (1.1) for
n = 1 and n = 2. I. Skau and E. Selmer[11] use a similarmethod,but they do
evaluateC(2n)in termsof Bernoullinumbers.
not explicitly
Apostol'spaper [1] containsa surveyof "elementary"methodsused to establish
(1.1). An evenmorerecentpaper of E. L. Stark[12] containsa lengthybibliography
cited in the two
of papers on the evaluationof C(2) and C(2n). To the references
papers,one can add the paper of R. Hovstad [3] and H. Radeaforementioned
macher'sbook [9, pp. 121-124].
In thispaper,two newproofsof (1.1) are given.The proofsuse only elementary
in an ordinarycalcalculus.The firstproof,especially,is suitableforpresentation
culus class.
The Bernoullipolynomials
2. Propertiesof Bernoullinumbersand polynomials.
<
by
defined
0
are
oo,
<
n
B.(x),
(2.1)
text
e' __ - =
a)
Bn(X)tnln!
2i)o
(|t I <