elementary number theory notes - santos

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1、ElementaryNumberTheoryNotescDavidA.SantosJanuary15,2004iiContentsPrefacev1Preliminaries11.1Introduction..........................11.2Well-Ordering.........................21.3MathematicalInduction...................41.4BinomialCoefficients.....................161.5Viete’sFormulæ`.......

2、................161.6FibonacciNumbers.....................161.7PigeonholePrinciple.....................232Divisibility312.1Divisibility............................312.2DivisionAlgorithm.......................342.3SomeAlgebraicIdentities..................383Congruences.Zn473.1Congruence

3、s.........................473.2DivisibilityTests.........................573.3CompleteResidues......................604UniqueFactorisation634.1GCDandLCM........................634.2Primes..............................734.3FundamentalTheoremofArithmetic...........76iiiivCONTENTS5LinearD

4、iophantineEquations895.1EuclideanAlgorithm.....................895.2LinearCongruences.....................945.3AtheoremofFrobenius...................965.4ChineseRemainderTheorem................1006Number-TheoreticFunctions1056.1GreatestIntegerFunction..................1056.2DePolignac’

5、sFormula....................1166.3ComplementarySequences................1196.4ArithmeticFunctions.....................1216.5Euler’sFunction.ReducedResidues............1286.6MultiplicationinZn......................1346.7MobiusFunction¨.......................1387MoreonCongruences1417.1Th

6、eoremsofFermatandWilson..............1417.2Euler’sTheorem........................1478ScalesofNotation1518.1TheDecimalScale......................1518.2Non-decimalScales.....................1578.3AtheoremofKummer....................1619DiophantineEquations1659.1MiscellaneousDiophantineeq

7、uations..........16510MiscellaneousExamplesandProblems16910.1MiscellaneousExamples...................17011PolynomialCongruences17312QuadraticReciprocity17513ContinuedFractions177PrefaceThesenotesstartedinthesummerof1993whenIwasteachingNumberTheoryattheCenterforTalentedYouthSummerProg