几个同余式猜想的证明

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1、DissertationforMasterofSciencePR00FS0FSOMECONJECTURESONCONGRUENCESQinGaoSupervisedbyProfessorZhiweiSunFundamentaIMathematicsDepartmentofMathematics，NanjingUniversityMay,2011黜㈣140㈣6Y237毕业论文题目：基础数学几个同余式猜想的证明专业2008级硕士生姓名：高勤指导教师(姓名、职称)：型：塑堡鳌丝摘要本文我们主要利用Berno

2、ulli数与调和数方法研究几个同余式猜想。对于素数P>3，我们证明了如下结果：(i)．f2惫+1)9知39p(2)(modp)，其中gp(2)是FerinatN(2p～一1)／p，这是级数∑善面i1矿=(3ln2)／2的同余式模拟。(ii)．厶p-1丽kT等三一三2(考)(modp)，厶f一31％一＼3／r⋯～川其中(一)是Legendre符号，中心三项式系数死是(z2+z+1)惫中zk项系数。(iii)．蓦尼(石})p一1三譬+譬cm。d矿，，且喜尼(一寡b)p+l三乞；一等cm。dp5，关键词：中

3、心三项式系数：超同余式；Bernoulli数；调和数。THESIS：PROOFSOFSOMECONJECTURESONCONGRUENCESSPECIALIZATION：FundamentalMathematicsPOSTGRADUATE：QinGaoMENTOR：ProfessorZhiweiSunAbstractInthispaper，weprovesomeconjecturesoncongruencesmoduloprimepowersviaBernoullinumbersandharmonic

4、numbers．ForanyprimeP>3，weshowthefollowingresults：f2忌+1)9南3qp(2)(modp)，whereqp(2)denotestheFermatquotient(2p--1__1)／p，thisisacongruenceanalogueoftheidentityEk一=0两醪1=(31n2)／2．(ii)．三一互1(詈)modp)，where(一)istheLegendresymbolandthecentraltrinomialcoefficient死i

5、sthecoefficientofz七in(X2+z+1)忌．and薹南(妻)p-1三pi2+pi3(modp4，，篆p-1七(一声)p+l三p百3一pi4(modp5，Keywords：centraltrinomialcoefficients；super--congruences；Bernoullinum．．bers；harmonicnumbers盟叫¨∑脚ContentsChineseAbstract．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．

6、···········1Abstract⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯·llChapter1Introduction⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯··1Chapter2ProofsofTheorems1．1and1．2⋯⋯⋯⋯⋯⋯⋯⋯⋯5Chapter3ProofofTheorem1．3⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯9Bibliography⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯18Acknowledgements⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯·19111Chapter1l

7、ntroductionCongruencesarefundamentaltopicsinnumbertheoryandcongruencesrood-uloprimeshavebeenwidelyinvestigatedsincethetimeofFermat．Therearemanyfamouscongruences．Forexample，Fermat’Slittletheorem([4]，Theoreml．1)：LetPbeanyprime．Thenforeachintegera，wehaveap

8、三a(modp)，inotherwords．ap_1三1(modP)foranyintegeranotdivisiblebyP．Wilson’Stheorem(【4]jTheoreml．3)：AnintegerP>1isaprimeifandonlyif∞一1)!三一1(modp)．Wolstenholme’Stheorem(see[2】)：ForanyprimeP>3．Wehave(翟二沪1(modp3)．Ithasbeenconjecturedtha