| fullrecord |
[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Paavola, Antti", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-08-09T05:21:16Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-08-09T05:21:16Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/65209", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielmassa etsit\u00e4\u00e4n ratkaisuja Pellin yht\u00e4l\u00f6lle eli muotoa x^2-Dy^2=1, jossa luku D ei saa olla mink\u00e4\u00e4n luvun neli\u00f6. Ty\u00f6ss\u00e4 keskityt\u00e4\u00e4n Pellin yht\u00e4l\u00f6n kokonaislukuratkaisujen l\u00f6yt\u00e4miseen, sill\u00e4 reaalilukuratkaisut l\u00f6ydet\u00e4\u00e4n helposti. Samoin helposti l\u00f6ydet\u00e4\u00e4n jokaisen Pellin yht\u00e4l\u00f6n toteuttavat triviaaliratkaisut. Aluksi tutustutaan kolmiolukuihin ja neli\u00f6lukuihin. Kolmioneli\u00f6luvut liittyv\u00e4t olennaisesti Pellin yht\u00e4l\u00f6n erikoistapaukseen, jossa D=2. Tutkielman alkupuoliskolla tutkitaan t\u00e4t\u00e4 erikoistapausta ja laajennetaan saadut tulokset koskemaan kaikkia Pellin yht\u00e4l\u00f6it\u00e4. Tutkielman alkupuolen merkitt\u00e4vin tulos on se, ett\u00e4 l\u00f6yt\u00e4m\u00e4ll\u00e4 Pellin yht\u00e4l\u00f6lle yhden ratkaisun, saadaan loput kyseisen yht\u00e4l\u00f6n ratkaisut ensimm\u00e4isest\u00e4 ratkaisusta potenssiin korottamisen avulla.\nTutkielman j\u00e4lkipuoliskolla keskityt\u00e4\u00e4n ketjumurtolukuihin, koska Pellin yht\u00e4l\u00f6n pienin ratkaisu l\u00f6ydet\u00e4\u00e4n niiden avulla. Tuon ratkaisun l\u00f6yt\u00e4mist\u00e4 varten tarvitaan konvergentin ja jaksollisen ketjumurtoluvun k\u00e4sitteet. L\u00e4hes jokainen luku voidaan esitt\u00e4\u00e4 ketjumurtolukuna ja pienimm\u00e4n ratkaisun l\u00f6yt\u00e4mist\u00e4 varten t\u00e4ytyy luvun D neli\u00f6juuri esitt\u00e4\u00e4 ketjumurtolukuna, jossa alkaa toistua tietty jakso. Tutkielman lopulla k\u00e4yd\u00e4\u00e4n l\u00e4pi t\u00e4rke\u00e4t kaavat, joiden avulla saadaan laskettua jaksoa ja konvergentteja hy\u00f6dynt\u00e4en Pellinyht\u00e4l\u00f6n pienin ratkaisu. Ratkaisu lasketaan eri kaavoilla riippuen siit\u00e4 onko luku D parillinen vai pariton. T\u00e4m\u00e4n ratkaisun avulla sitten saadaan laskettua kaikki loput Pellin yht\u00e4l\u00f6n ratkaisut.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-08-09T05:21:16Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-08-09T05:21:16Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "46", "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": "Pellin yht\u00e4l\u00f6ist\u00e4", "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-201908093795", "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": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukuteoria", "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"}]
|