Selecta. 2 Vols: Elementary, analytic and geometric number theory: I,II (2 Vols.) PDF
Preview Selecta. 2 Vols: Elementary, analytic and geometric number theory: I,II (2 Vols.)
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:+41 (0)44 632 34 36 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
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