Solving Problems with Magma.pdf

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1、SolvingProblemswithMagmaWiebBosmaJohnCannonCatherinePlayoustAllanSteelSchoolofMathematicsandStatisticsUniversityofSydneyNSW2006,AustraliaCopyright°c1999.Allrightsreserved.Nopartofthisbookmaybereproducedwithoutwrittenpermission.TypesetbycomputerusingDonaldKnuth’sT

2、EX,andthedocumentpreparationsystemLATEXdevelopedbyLeslieLamport.IntroductionThisbookisneitheranintroductorymanualnorareferencemanualforMagma.ThoseneedsaremetbythebooksAnIntroductiontoMagmaandHandbookofMagmaFunctions.Eventhemostkeeninductivelearnerswillnotlearnall

3、thereistoknowaboutMagmafromthepresentwork.WhatSolvingProblemswithMagmadoesofferisalargecollectionofreal-worldalgebraicproblems,solvedusingtheMagmalanguageandintrinsics.Itishopedthatbystudyingtheseexamples,especiallythoseinyourspecialty,youwillgainapracticalunderst

4、andingofhowtoexpressmath-ematicalproblemsinMagmaterms.Mostoftheexampleshavearisenfromgenuineresearchquestions,someofwhichtheothercomputeralgebrasystemsonthemarketcannothandlewell,andafewwhichstretchMagmatoitslimittoo.IfyouaretryingtheexamplesonyourownMagma,bewarn

5、edthatsomeoftheexamplesrequiresignificantCPUtimeand/orstoragespacetocompletetheirexecution.Wethankallthosewhohavecontributedexamples.Someareolderproblems,solvedwithMagma’spredecessorCayley.OthersarenewonesthatexploitMagma’sabilitytostraddleallaspectsofalgebra.Wewe

6、lcomecommentsonthisbookandsubmissionsofnewapproachesthatyouthinkmightadvantageothermathematicians.iiiContentsIntroductioniii1Language11.1Puzzle-solving:::::::::::::::::::::::::::::::::::::::11.1.1DogDaze:::::::::::::::::::::::::::::::::::::11.1.2Lettersstandingfo

7、rdigits::::::::::::::::::::::::::::21.2Sets,sequencesandfunctions::::::::::::::::::::::::::::::51.2.1Fareysequence::::::::::::::::::::::::::::::::::51.2.2Theknapsackproblem::::::::::::::::::::::::::::::71.2.3Simulationofacellularautomaton:::::::::::::::::::::::72

8、TheIntegers112.1Introduction::::::::::::::::::::::::::::::::::::::::112.2ArithmeticandArithmeticFunctions::::::::::::::::::::::::::112.2.1Example:Amicablenumbe