Diophantine Analysis (Hardcover)

INTRODUCTION: BASIC PRINCIPLESWho was Diophantus?Pythagorean triplesFermat's last theoremThe method of infinite descentCantor's paradiseIrrationality of eIrrationality of pi Approximating with rationalsLinear diophantine equationsExercisesCLASSICAL APPROXIMATION THEOREMSDirichlet's approximation theoremA first irrationality criterionThe order of approximationKronecker's approximation theoremBilliardUniform distributionThe Farey sequenceMediants and Ford circlesHurwitz' theoremPade approximationExercisesCONTINUED FRACTIONSThe Euclidean algorithm revisited and calendarsFinite continued fractionsInterlude: Egyptian fractionsInfinite continued fractionsApproximating with convergentsThe law of best approximationsConsecutive convergentsThe continued fraction for eExercisesTHE IRRATIONALITY OF z(3)The Riemann zeta-functionApery's theoremApproximating z(3)A recursion formulaThe speed of convergenceFinal steps in the proofAn irrationality measureA non-simple continued fractionBeukers' proofNotes on recent resultsExercisesQUADRATIC IRRATIONALSFibonacci numbers and paper foldingPeriodic continued fractionsGalois' theoremSquare rootsEquivalent numbersSerret's theoremThe Marko® spectrumBadly approximable numbersNotes on the metric theoryExercisesTHE PELL EQUATIONThe cattle problemLattice points on hyperbolasAn infinitude of solutionsThe minimal solutionThe group of solutionsThe minus equationThe polynomial Pell equationNathanson's theoremNotes for further readingExercisesFACTORING WITH CONTINUED FRACTIONSThe RSA cryptosystemA diophantine attack on RSAAn old idea of FermatCFRACExamples of failuresWeighted mediants and a refinementNotes on primality testingExercisesGEOMETRY OF NUMBERSMinkowski's convex body theoremGeneral latticesThe lattice basis theoremSums of squaresApplications to linear and quadratic formsThe shortest lattice vector problemGram-Schmidt and consequencesLattice reduction in higher dimensionsThe LLL-algorithmThe small integer problemNotes on sphere packingsExercisesTRANSCENDENTAL NUMBERSAlgebraic vs. transcendentalLiouville's theoremLiouville numbersThe transcendence of eThe transcendence of piSquaring the circle?Notes on transcendental numbersExercisesTHE THEOREM OF ROTHRoth's theoremThue equationsFinite vs. infiniteDifferential operators and indicesOutline of Roth's methodSiegel's lemmaThe index theoremWronskians and Roth's lemmaFinal steps in Roth's proofNotes for further readingExercisesTHE ABC-CONJECTUREHilbert's tenth problemThe ABC-theorem for polynomialsFermat's last theorem for polynomialsThe polynomial Pell equation revisitedThe abc-conjectureLLL & abcThe ErdAos-Woods conjectureFermat, Catalan & co.Mordell's conjectureNotes on abcExercisesP-ADIC NUMBERSNon-Archimedean valuationsUltrametric topologyOstrowski's theoremCurious convergenceCharacterizing rationalsCompletions of the rationalsp-adic numbers as power seriesError-free computingNotes on the p-adic interpolation of the zeta-functionExercisesHENSEL'S LEMMA AND APPLICATIONSp-adic integersSolving equations in p-adic numbersHensel's lemmaUnits and squaresRoots of unityHensel's lemma revisitedHensel lifting: factoring polynomialsNotes on p-adics: what we leave outExercisesTHE LOCAL-GLOBAL PRINCIPLEOne for all and all for oneThe theorem of Hasse-MinkowskiTernary quadraticsThe theorems of Chevalley and WarningApplications and limitationsThe local Fermat problemExercisesAPPENDIX: ALGEBRA AND NUMBER THEORYGroups, rings, and fieldsPrime numbersRiemann's hypothesisModular arithmeticQuadratic residuesPolynomialsAlgebraic number fieldsKummer's work on Fermat's last theoremBIBLIOGRAPHYINDEX