Selecta. 2 Vols: Elementary, analytic and geometric number theory: I,II (2 Vols.)

545 Pages·2007·3.45 MB·English

by  Andrzej Schnizel

schinzel_vol_2_titelei 5.3.2007 18:34 Uhr Seite 1
H e r i t a g e o f E u r o p e a n M a t h e m a t i c s
Advisory Board
Michèle Audin, Strasbourg
Ciro Ciliberto, Roma
Ildar A. Ibragimov, St. Petersburg
Wladyslaw Narkiewicz, Wroclaw
Peter M. Neumann, Oxford
Samuel J. Patterson, Göttingen
schinzel_vol_2_titelei 5.3.2007 18:34 Uhr Seite 2
Andrzej Schinzel in 2007
schinzel_vol_2_titelei 5.3.2007 18:34 Uhr Seite 3
Andrzej Schinzel
Selecta
Volume II
Elementary, Analytic
and Geometric Number Theory
Edited by
Henryk Iwaniec
Wladyslaw Narkiewicz
Jerzy Urbanowicz
schinzel_vol_2_titelei 5.3.2007 18:34 Uhr Seite 4
Author:
Andrzej Schinzel
Institute of Mathematics
Polish Academy of Sciences
ul.Śniadeckich 8,skr.poczt.21
00-956 Warszawa 10
Poland
Editors:
Henryk Iwaniec Władysław Narkiewicz Jerzy Urbanowicz
Department of Mathematics Institute of Mathematics Institute of Mathematics
Rutgers University University of Wrocław Polish Academy of Sciences
New Brunswick,NJ 08903 pl.Grunwaldzki 2/4 ul.Śniadeckich 8,skr.poczt.21
U.S.A. 50-384 Wrocław 00-956 Warszawa 10
[email protected] Poland Poland
[email protected] [email protected]
2000 Mathematics Subject Classification:11,12
ISBN 978-3-03719-038-8 (Set Vol I & Vol II)
The Swiss National Library lists this publication in The Swiss Book,the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright.All rights are reserved,whether the whole or part of the material is
concerned,specifically the rights of translation,reprinting,re-use of illustrations,recitation,broadca-
sting,reproduction on microfilms or in other ways,and storage in data banks.For any kind of use per-
mission of the copyright owner must be obtained.
© 2007 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum FLI C4
CH-8092 Zürich
Switzerland
Phone:[phone]
Email:[email protected]
Homepage:www.ems-ph.org
Printed in Germany
9 8 7 6 5 4 3 2 1
Contents
Volume 2
G. Arithmeticfunctions 859
CommentaryonG:Arithmeticfunctions
byKevinFord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861
G1 Onfunctionsϕ(n)andσ(n) . . . . . . . . . . . . . . . . . . . . . . . . . 866
G2 Surl’équationϕ(x)=m . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
G3 Surunproblèmeconcernantlafonctionϕ(n) . . . . . . . . . . . . . . . . 875
G4 Distributionsofthevaluesofsomearithmeticalfunctions
withP.Erdo˝s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
G5 Onthefunctionsϕ(n)andσ(n)
withA.Makowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890
(cid:4)
G6 Onintegersnotoftheformn−ϕ(n)
withJ.Browkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
H. Divisibilityandcongruences 899
CommentaryonH:Divisibilityandcongruences
byH.W.Lenstrajr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
H1 SurunproblèmedeP.Erdo˝s . . . . . . . . . . . . . . . . . . . . . . . . . 903
H2 Onthecongruenceax ≡b(modp) . . . . . . . . . . . . . . . . . . . . . 909
H3 Onthecompositeintegersoftheformc(ak+b)!±1 . . . . . . . . . . . 912
H4 Onpowerresiduesandexponentialcongruences . . . . . . . . . . . . . . 915
H5 Abelianbinomials,powerresiduesandexponentialcongruences . . . . . . 939
H6 AnextensionofWilson’stheorem
withG.Baron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
H7 Systemsofexponentialcongruences . . . . . . . . . . . . . . . . . . . . 975
H8 Onaprobleminelementarynumbertheory
withJ.Wójcik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987
H9 Onexponentialcongruences . . . . . . . . . . . . . . . . . . . . . . . . . 996
H10 Unecaractérisationarithmétiquedesuitesrécurrenteslinéaires
avecDanielBarskyetJean-PaulBézivin. . . . . . . . . . . . . . . . . 1001
H11 Onpowerresidues
withM.Skałba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012
vi Contents
I. Primitivedivisors 1031
CommentaryonI:Primitivedivisors
byC.L.Stewart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033
I1 Onprimitiveprimefactorsofan−bn . . . . . . . . . . . . . . . . . . . . 1036
I2 OnprimitiveprimefactorsofLehmernumbersI . . . . . . . . . . . . . . 1046
I3 OnprimitiveprimefactorsofLehmernumbersII . . . . . . . . . . . . . . 1059
I4 OnprimitiveprimefactorsofLehmernumbersIII . . . . . . . . . . . . . 1066
I5 PrimitivedivisorsoftheexpressionAn−Bninalgebraicnumberfields . . 1090
I6 Anextensionofthetheoremonprimitivedivisorsinalgebraicnumberfields 1098
J. Primenumbers 1103
CommentaryonJ:Primenumbers
byJerzyKaczorowski . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
J1 Surcertaineshypothèsesconcernantlesnombrespremiers
withW.Sierpin´ski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113
J2 Remarksonthepaper“Surcertaineshypothèsesconcernantlesnombres
premiers” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134
J3 AremarkonapaperofBatemanandHorn . . . . . . . . . . . . . . . . . 1142
J4 OntwotheoremsofGelfondandsomeoftheirapplications
Section5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145
J5 Ontherelationbetweentwoconjecturesonpolynomials . . . . . . . . . . 1154
K. Analyticnumbertheory 1193
CommentaryonK:Analyticnumbertheory
byJerzyKaczorowski . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195
K1 OnSiegel’szero
withD.M.Goldfeld . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199
K2 Multiplicativepropertiesofthepartitionfunction
withE.Wirsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211
K3 OnananalyticproblemconsideredbySierpin´skiandRamanujan . . . . . 1217
K4 ClassnumbersandshortsumsofKroneckersymbols
withJ.UrbanowiczandP.VanWamelen . . . . . . . . . . . . . . . . . 1224
L. Geometryofnumbers 1245
CommentaryonL:Geometryofnumbers
byWolfgangM.Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . 1247
L1 AdecompositionofintegervectorsII
withS.Chaładus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249
L2 AdecompositionofintegervectorsIV . . . . . . . . . . . . . . . . . . . 1259
L3 ApropertyofpolynomialswithanapplicationtoSiegel’slemma . . . . . 1274
L4 Onvectorswhosespancontainsagivenlinearsubspace
withI.AlievandW.M.Schmidt . . . . . . . . . . . . . . . . . . . . . 1288
Contents vii
M. Otherpapers 1303
CommentaryonM:Otherpapers
byStanisławKwapien´ . . . . . . . . . . . . . . . . . . . . . . . . . . 1305
TheinfluenceoftheDavenport–Schinzelpaperindiscrete
andcomputationalgeometry
byEndreSzemerédi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311
M1 Surl’équationfonctionnellef[x+y·f(x)]=f(x)·f(y)
avecS.Gołab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314
(cid:4)
M2 Acombinatorialproblemconnectedwithdifferentialequations
withH.Davenport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327
M3 AnanalogueofHarnack’sinequalityfordiscretesuperharmonic
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338
M4 Aninequalityfordeterminantswithrealentries . . . . . . . . . . . . . . . 1347
M5 ComparisonofL1-andL∞-normsofsquaresofpolynomials
withW.M.Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1350
Unsolvedproblemsandunprovedconjectures 1365
Unsolvedproblemsandunprovedconjecturesproposedby
AndrzejSchinzelintheyears1956–2006arrangedchronologically . . . . 1367
PublicationlistofAndrzejSchinzel 1375
viii Contents
Volume 1
A. Diophantineequationsandintegralforms 1
CommentaryonA:Diophantineequationsandintegralforms
byR.Tijdeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A1 SurlesnombresdeMersennequisonttriangulaires
avecGeorgesBrowkin . . . . . . . . . . . . . . . . . . . . . . . . . . 11
A2 Surquelquespropriétésdesnombres3/net4/n,oùnestunnombreimpair 13
A3 Surl’existenced’uncerclepassantparunnombredonnédepointsaux
coordonnéesentières . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A4 Surlessommesdetroiscarrés . . . . . . . . . . . . . . . . . . . . . . . . 18
(cid:2)
A5 OntheDiophantineequation n A xϑk =0 . . . . . . . . . . . . . . . 22
k=1 k k
A6 Polynomialsofcertainspecialtypes
withH.DavenportandD.J.Lewis . . . . . . . . . . . . . . . . . . . . 27
A7 AnimprovementofRunge’stheoremonDiophantineequations . . . . . . 36
A8 Ontheequationym =P(x)
withR.Tijdeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A9 Zetafunctionsandtheequivalenceofintegralforms
withR.Perlis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A10 QuadraticDiophantineequationswithparameters
withD.J.Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A11 Selmer’sconjectureandfamiliesofellipticcurves
withJ.W.S.Cassels . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A12 Familiesofcurveshavingeachanintegerpoint . . . . . . . . . . . . . . . 67
A13 Hasse’sprincipleforsystemsofternaryquadraticformsandforone
biquadraticform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A14 OnRunge’stheoremaboutDiophantineequations
withA.Grytczuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A15 Onsumsofthreeunitfractionswithpolynomialdenominators . . . . . . . 116
A16 Onequationsy2 =xn+kinafinitefield
withM.Skałba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B. Continuedfractions 127
CommentaryonB:Continuedfractions
byEugèneDubois . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B1 Onsomeproblemsofthearithmeticaltheoryofcontinuedfractions . . . . 131
B2 OnsomeproblemsofthearithmeticaltheoryofcontinuedfractionsII . . . 149
B3 OntwoconjecturesofP.ChowlaandS.Chowlaconcerningcontinued
fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Contents ix
C. Algebraicnumbertheory 167
CommentaryonC:Algebraicnumbers
byDavidW.BoydandD.J.Lewis . . . . . . . . . . . . . . . . . . . . 169
C1 ArefinementoftwotheoremsofKronecker
withH.Zassenhaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
C2 OnatheoremofBauerandsomeofitsapplications . . . . . . . . . . . . 179
C3 AnextensionofthetheoremofBauerandpolynomialsofcertainspecialtypes
withD.J.LewisandH.Zassenhaus . . . . . . . . . . . . . . . . . . . 190
C4 Onsumsofrootsofunity.(SolutionoftwoproblemsofR.M.Robinson) . 197
C5 OnatheoremofBauerandsomeofitsapplicationsII . . . . . . . . . . . 210
C6 Ontheproductoftheconjugatesoutsidetheunitcircleofanalgebraicnumber 221
C7 Onlineardependenceofroots . . . . . . . . . . . . . . . . . . . . . . . . 238
C8 OnSylow2-subgroupsofK O forquadraticnumberfieldsF
2 F
withJ.Browkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
C9 Aclassofalgebraicnumbers . . . . . . . . . . . . . . . . . . . . . . . . 264
C10 OnvaluesoftheMahlermeasureinaquadraticfield(solutionofaproblem
ofDixonandDubickas) . . . . . . . . . . . . . . . . . . . . . . . . . 272
D. Polynomialsinonevariable 281
CommentaryonD:Polynomialsinonevariable
byMichaelFilaseta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
D1 Solutiond’unproblèmedeK.Zarankiewiczsurlessuitesdepuissances
consécutivesdenombresirrationnels . . . . . . . . . . . . . . . . . . 295
D2 Onthereducibilityofpolynomialsandinparticularoftrinomials . . . . . 301
D3 Reducibilityofpolynomialsandcoveringsystemsofcongruences . . . . . 333
D4 ReducibilityoflacunarypolynomialsI . . . . . . . . . . . . . . . . . . . 344
D5 ReducibilityoflacunarypolynomialsII . . . . . . . . . . . . . . . . . . . 381
D6 Anoteonthepaper“ReducibilityoflacunarypolynomialsI”
withJ.Wójcik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
D7 ReducibilityoflacunarypolynomialsIII . . . . . . . . . . . . . . . . . . 409
D8 ReducibilityoflacunarypolynomialsIV . . . . . . . . . . . . . . . . . . 447
D9 Onthenumberoftermsofapowerofapolynomial . . . . . . . . . . . . 450
D10 Onreducibletrinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
D11 OnaconjectureofPosnerandRumsey
withK.Gyo˝ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
D12 ReducibilityoflacunarypolynomialsXII . . . . . . . . . . . . . . . . . . 563
D13 OnreducibletrinomialsII . . . . . . . . . . . . . . . . . . . . . . . . . . 580
D14 OnreducibletrinomialsIII . . . . . . . . . . . . . . . . . . . . . . . . . . 605
D15 OnthegreatestcommondivisoroftwounivariatepolynomialsI . . . . . . 632
D16 OnthegreatestcommondivisoroftwounivariatepolynomialsII . . . . . 646
D17 Onthereducedlengthofapolynomialwithrealcoefficients . . . . . . . . 658
x Contents
E. Polynomialsinseveralvariables 693
CommentaryonE:Polynomialsinseveralvariables
byUmbertoZannier . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
E1 Someunsolvedproblemsonpolynomials . . . . . . . . . . . . . . . . . . 703
E2 Reducibilityofpolynomialsinseveralvariables . . . . . . . . . . . . . . 709
E3 Reducibilityofpolynomialsoftheformf(x)−g(y) . . . . . . . . . . . 715
E4 Reducibilityofquadrinomials
withM.Fried . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
E5 Ageneralirreducibilitycriterion . . . . . . . . . . . . . . . . . . . . . . . 739
E6 Somearithmeticpropertiesofpolynomialsinseveralvariables
withH.L.Montgomery . . . . . . . . . . . . . . . . . . . . . . . . . . 747
E7 Ondifferencepolynomialsandhereditarilyirreduciblepolynomials
withL.A.RubelandH.Tverberg . . . . . . . . . . . . . . . . . . . . 755
E8 Onadecompositionofpolynomialsinseveralvariables . . . . . . . . . . 760
E9 Onweakautomorphsofbinaryformsoveranarbitraryfield . . . . . . . . 779
E10 Reducibilityofsymmetricpolynomials . . . . . . . . . . . . . . . . . . . 828
F. Hilbert’sIrreducibilityTheorem 835
CommentaryonF:Hilbert’sIrreducibilityTheorem
byUmbertoZannier . . . . . . . . . . . . . . . . . . . . . . . . . . . 837
F1 OnHilbert’sIrreducibilityTheorem . . . . . . . . . . . . . . . . . . . . . 839
F2 Aclassofpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846
F3 TheleastadmissiblevalueoftheparameterinHilbert’sIrreducibility
Theorem
withUmbertoZannier . . . . . . . . . . . . . . . . . . . . . . . . . . 849