Fermatovi brojevi

master's thesis

• Author: Ivana Lepan
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Fermatovi brojevi su brojevi oblika \(f_n = 2^{2n} + 1, n = 0, 1, 2, \dots\) Ukoliko je \(f_n\) prost broj, nazivamo ga Fermatovim prostim brojem. Fermat je smatrao da su svi brojevi tog oblika prosti brojevi, no Euler je opovrgnuo tu tvrdnju pokazavši da je Fermatov broj \(f_5\) složen broj. Stoga se možemo pitati postoji li beskonačno mnogo prostih brojeva oblika \(2^{2n} + 1\). Hipoteza je da je samo konačno mnogo Fermatovih brojeva prosto. No, to je još uvijek otvoren, ali ne i jedini, problem na koji nismo odgovorili. Štoviše, proučavanje Fermatovih brojeva otvorilo je mnogobrojne probleme koje matematičari još uvijek pokušavaju razriješiti. U radu su navedeni neki od najpoznatijih otvorenih problema. No, naglasak ovog rada nije na otvorenim problemima, nego na rezultatima do kojih su došli poznati matematičari proučavajući Fermatove brojeve. Stoga u ovom radu iznosimo neka osnovna svojstva Fermatovih brojeva, dajemo testove prostosti za Fermatove brojeve i opisujemo djelitelje složenih Fermatovih brojeva.

Abstract (english)

Numbers of the form \(f_n = 2^{2n} + 1, n = 0, 1, 2, \dots\) are called Fermat numbers. Prime Fermat numbers are called Fermat primes. Fermat conjectured that all numbers of that form are prime. However, Euler disproved the Fermat's conjecture when he showed that \(f_5\) is composite. So, the question is: Are there infinitely many prime numbers of the form \(2^{2n} + 1, n \in \mathbb{N}\)? The conjecture states that the set of prime Fermat numbers is finite. However, that is an open problem to which we haven't found an answer. Furthermore, the study of Fermat numbers has opened many other problems which are yet to be solved. In this paper we have named some of the well known ones. The aim of this work is not to put the emphasis on open problems, but on the results found by many well known mathematicians while studying Fermat numbers. So, in this work we consider some basic properties of Fermat numbers, give some primality tests for Fermat numbers and describe factors of composite Fermat numbers.

• Keywords: Fermatovi brojevi; Fermatov prosti broj; Euler; testovi prostosti; djelitelji složenih Fermatovih brojeva
• Keywords (english): Fermat numbers; prime Fermat numbers; Euler; primality tests; factors of composite Fermat numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:404005
• 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: Access restricted to students and staff of home institution
• Created on: 2019-02-22 11:09:49