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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "V\u00e4is\u00e4nen, Jussi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-01-15T11:51:41Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-01-15T11:51:41Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/62510", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on perehty\u00e4 Fermat'n suuren lauseen todistuksen syntyyn ja etenkin muutamiin lauseen yksinkertaisimpiin erityistapauksiin. Fermat'n suuren lauseen mukaan ei ole olemassa kokonaislukuja x, y ja z, jotka toteuttavat yht\u00e4l\u00f6n x^n + y^n = z^n,\n\nkun n on lukua 2 suurempi luonnollinen luku. Vaikka lause on nimetty 1600-luvulla el\u00e4neen Pierre de Fermat'n mukaan, ulottuvat sen juuret tuhansien vuosien p\u00e4\u00e4h\u00e4n Fermat'ta edelt\u00e4v\u00e4\u00e4n aikaan. Fermat'n suuri lause onnistuttiin my\u00f6s lopulta todistamaan vasta satojen vuosien kuluttua siit\u00e4, kun Fermat oli t\u00e4m\u00e4n v\u00e4itt\u00e4m\u00e4n esitt\u00e4nyt. Andrew Wiles yhdisti lopullisessa todistuksessa onnistuneesti vuosisatojen varrella kehittyneit\u00e4 tuloksia monilta eri matematiikan aloilta ja lauseen todistaminen vaati h\u00e4nelt\u00e4 seitsem\u00e4n vuoden yht\u00e4jaksoisen ty\u00f6n.\n\n\n\nT\u00e4ss\u00e4 tutkielmassa otetaan katsaus Fermat'n suuren lauseen historiaan ja todistetaan lauseen paikkaansapit\u00e4vyys tapauksissa n=4 ja n=3. Tapauksen n=4 todistus pohjautuu jo Fermat'n k\u00e4ytt\u00e4m\u00e4\u00e4n \u00e4\u00e4rett\u00f6m\u00e4n laskeutumisen menetelm\u00e4\u00e4n, kun taas tapaus n=3 on todistettu Eulerin laatiman todistuksen pohjalta.\n\n\n\nEulerin todistuksessa tapaukselle n=3 hy\u00f6dynnet\u00e4\u00e4n Gaussin resiprookkilakia. Jos p ja q ovat erisuuria parittomia alkulukuja ja tiedet\u00e4\u00e4n, onko q neli\u00f6nj\u00e4\u00e4nn\u00f6s vai neli\u00f6nep\u00e4j\u00e4\u00e4nn\u00f6s modulo p, niin Gaussin resiprookkilaki kertoo, onko p t\u00e4ll\u00f6in neli\u00f6nj\u00e4\u00e4nn\u00f6s vai neli\u00f6nep\u00e4j\u00e4\u00e4nn\u00f6s modulo q. T\u00e4m\u00e4 resiprookkilain sis\u00e4lt\u00f6 saadaan esitetty\u00e4 suoraviivaisemmin Legendren symbolia hy\u00f6dynt\u00e4en ja ennen lain todistamista todistetaan aputuloksina muun muassa Eulerin kriteeri sek\u00e4 Gaussin lemma.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-01-15T11:51:41Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-01-15T11:51:41Z (GMT). No. of bitstreams: 0\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "41", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Fermat'n suuren lauseen erikoistapauksia", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201901151197", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "alkuluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lauseet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukuteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Fermatin suuri lause", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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