Fermatov doprinos u teoriji brojeva

master's thesis

• Author: Valentina Plantak
• Mentor: Zrinka Franušić (mentor)
• Committee member: Goran Radunović (predsjednik povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pierre de Fermat jedan je od najvećih matematičara 17. stoljeća. Smatra ga se utemeljiteljem moderne teorije brojava. Glavni cilj ovog diplomskog rada je prikazati i dijelom dokazati Fermatove rezultate i doprinose u teoriji brojeva. Fermat je dao čitav niz tvrdnji, uglavnom bez dokaza, koje su se godinama, a neke i stoljećima kasnije pokazale istinitima. Fermat je poznat po brojevima oblika \(F_m=2^{2^{m}}+1, m \in \mathbb{N}\) koji se zovu Fermatovi brojevi te po svom doprinosu u razvoju diofantskih jednadžbi. U radu su iskazani i dokazani (svi osim posljednjeg) Fermatovi teoremi: Mali Fermatov teorem, Teorem o zbroju dva kvadrata, Teorem o poligonalnim brojevima i Veliki Fermatov teorem. Osim teorema u radu su opisane i na primjerima objašnjene dvije Fermatove metode: Fermatova metoda faktorizacije i Fermatova metoda neprekidnih silazaka.

Abstract (english)

Pierre de Fermat is one of the greatest mathematicians of 17. century. He is considered a founder of modern number theory. The main purpose of this work is to present and partly to prove the results and contributions Fermat’s contributions to number theory. Fermat gave a whole series of assertions, mostly without the proof, which have been years, and some of them centuries later proved to be true. Fermat is famous for his numbers \(F_m=2^{2^{m}}+1, m \in \mathbb{N}\), which we call Fermat’s numbers, and for his contributions in development of Diophant’s equations. In the work are presented and proved (all except the last one) following theorems: Fermat’s Little theorem, Two-Square Theorem, Fermat polygonal number theorem and Fermat’s Last Theorem. Except the theorems, two Fermat’s methods: Fermat’s factorizations method and Fermat’s method of infinite descent are described and explained trough the examples.

• Keywords: Fermat; teorija brojeva; Fermatovi brojevi; diofantske jednadžbe; Mali Fermatov teorem; Teorem o zbroju dva kvadrata; Teorem o poligonalnim brojevima; Veliki Fermatov teorem; Fermatova metoda faktorizacije; Fermatova metoda neprekidnih silazaka
• Keywords (english): Fermat; number theory; Fermat’s numbers; Diophant’s equations; Fermat’s Little theorem; Two-Square Theorem; Fermat polygonal number theorem; Fermat’s Last Theorem; Fermat’s factorizations method; Fermat’s method of infinite descent
• Language: croatian
• URN:NBN: urn:nbn:hr:217:946216
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-06-10 12:16:01