Fermatov doprinos u teoriji brojeva

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.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