Number theory structures examples and problems.pdf

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1、i”Godmadetheintegers,allelseistheworkofman.”LeopoldKroneckeriiNUMBERTHEORYStructures,Examples,andProblemsTituAndreescuDorinAndricaiiContentsForeword7Acknowledgments9Notation11ISTRUCTURES,EXAMPLES,ANDPROBLEMS131Divisibility151.1Divisibility...........................151

2、.2Primenumbers.........................211.3Thegreatestcommondivisor.................301.4Oddandeven..........................391.5Modulararithmetics......................421.6Chineseremaindertheorem..................471.7Numericalsystems.......................501.7.1R

3、epresentationofintegersinanarbitrarybase...501.7.2Divisibilitycriteriainthedecimalsystem......512Contents2PowersofIntegers612.1Perfectsquares.........................612.2Perfectcubes..........................702.3kthpowersofintegers,k≥4..................723FloorFuncti

4、onandFractionalPart773.1Generalproblems........................773.2Floorfunctionandintegerpoints...............833.3Anusefulresult.........................884DigitsofNumbers914.1Thelastdigitsofanumber..................914.2Thesumofthedigitsofanumber..............944.3

5、Otherproblemsinvolvingdigits................1005BasicPrinciplesinNumberTheory1035.1Twosimpleprinciples.....................1035.1.1Extremalarguments..................1035.1.2Pigeonholeprinciple..................1055.2Mathematicalinduction....................1085.3Infini

6、tedescent.........................1135.4Inclusion-exclusion.......................1156ArithmeticFunctions1196.1Multiplicativefunctions....................1196.2Numberofdivisors.......................1266.3Sumofdivisors.........................1296.4Euler’stotientfuncti

7、on.....................1316.5ExponentofaprimeandLegendre’sformula........1357MoreonDivisibility1417.1Fermat’sLittleTheorem....................1417.2Euler’sTheorem........................1477.3Theorderofanelement....................1507.4Wilson’sTheorem.................

8、......1538DiophantineEquations1578.1LinearDiophantineequations.................1578.2QuadraticDiophantineequations...