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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "M\u00f6rsky, Matti", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-06T07:16:22Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-06T07:16:22Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95581", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 ty\u00f6ss\u00e4 tutkitaan primitiivisi\u00e4 juuria ja niiden erilaisia sovelluksia. Sovelluksissa k\u00e4yd\u00e4\u00e4n l\u00e4pi rationaalilukujen desimaaliesityksen ominaisuuksia, n\u00e4enn\u00e4issatunnauslukugeneraattorin teoriaa ja indeksiaritmetiikkaa.\n\nAluksi ty\u00f6ss\u00e4 k\u00e4yd\u00e4\u00e4n l\u00e4pi olennaisia m\u00e4\u00e4ritelmi\u00e4, esimerkkej\u00e4 ja lauseita, joiden j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n primitiivinen juuri. Primitiivinen juuri on yksik\u00f6iden ryhm\u00e4n Un viritt\u00e4v\u00e4 alkio. T\u00e4ll\u00f6in yksik\u00f6iden ryhm\u00e4 on syklinen. Hy\u00f6dyllisen apulauseen avulla voidaan testata, onko jokin alkio primitiivinen juuri. Seuraavaksi tutkitaan syklisi\u00e4 yksik\u00f6iden ryhmi\u00e4 ja todistetaan, ett\u00e4 alkuluvuille p l\u00f6ytyy aina primitiivinen juuri yksik\u00f6iden ryhm\u00e4st\u00e4 Up. Lis\u00e4ksi huomataan syklisten ryhmien v\u00e4lisi\u00e4 yhteyksi\u00e4 ja m\u00e4\u00e4ritet\u00e4\u00e4n primitiivisten juurien m\u00e4\u00e4r\u00e4 Eulerin funktion avulla.\n\nPrimitiivisten juurten sovelluksissa tutkitaan rationaalilukujen desimaaliesityksien jaksollisuutta. Ensin selvitet\u00e4\u00e4n, milloin rationaaliluku on p\u00e4\u00e4ttyv\u00e4. Huomataan, ett\u00e4 jos luvun 1/n desimaaliesitys on p\u00e4\u00e4ttym\u00e4t\u00f6n, niin se on jaksollinen ja sen jakson pituus on korkeintaan n \u2212 1. Lis\u00e4ksi, jos 10 on primitiivinen juuri modulo n, niin desimaaliesityksen pituus on kertaluku \u03d5(n). \n\nT\u00e4m\u00e4n j\u00e4lkeen tarkastellaan n\u00e4enn\u00e4issatunnaislukuja tuottavaa menetelm\u00e4\u00e4 ja primitiivisten juurten k\u00e4ytt\u00f6\u00e4 n\u00e4enn\u00e4issatunnaisgeneraattorissa. Hy\u00f6dyllisen lauseen avulla voidaan l\u00f6yt\u00e4\u00e4 primitiivisen juuren potensseista viel\u00e4\nsuurempia primitiivisi\u00e4 juuria, jolloin n\u00e4enn\u00e4issatunnaislukujen l\u00f6yt\u00e4minen vaikeutuu. Lopuksi hy\u00f6dynnet\u00e4\u00e4n viel\u00e4 primitiivisi\u00e4 juuria m\u00e4\u00e4rittelem\u00e4\u00e4n indeksi, jota voidaan hy\u00f6dynt\u00e4\u00e4 kongruenssiyht\u00e4l\u00f6iden ratkaisemisessa sek\u00e4 m\u00e4\u00e4ritt\u00e4ess\u00e4 onko kongruenssiyht\u00e4l\u00f6ll\u00e4 ratkaisua ja kuinka monta niit\u00e4 on.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-06-06T07:16:22Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-06T07:16:22Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "42", "language": "", "element": "format", "qualifier": "extent", "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.subject.other", "value": "primitiivinen juuri", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yksik\u00f6iden ryhm\u00e4t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Primitiiviset juuret ja niiden sovellukset", "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-202406064341", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "rationaaliluvut", "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": "ryhm\u00e4t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "aritmetiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "alkuluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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