Savršeni brojevi

master's thesis

• Title (english): Perfect numbers
• Author: Josipa Atlija
• Mentor: Marija Galić (mentor)
• Committee member: Marija Galić (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Ivana Šain Glibić (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-09-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Savršeni brojevi interesantan su dio teorije brojeva. Matematičari ih godinama istražuju, no još uvijek nije poznata njihova primjena. Usko vezani uz savršene brojeve su i Mersenneovi brojevi. Jedan od načina pronalaska savršenih brojeva je pronalazak nešto jednostavnijih Mersenneovih prostih brojeva. U ovom diplomskom radu navedeni su matematički temelji potrebni za razumijevanje istoga. Nakon toga slijede definicija i svojstva Mersenneovih brojeva te se spominje i opisuje Lucas--Lehmerov test provjere prostosti. Naposljetku se govori o savršenim brojecima, njihovim svojstvima, višestruko savršenim brojevima te uvjetima koje bi neparni savršeni brojevi trebali zadovoljavati ukoliko postoje.

Abstract (english)

Perfect numbers are an interesting part of number theory. Mathematicians have been researching them for years, but their application is still unknown. Closely related to perfect numbers are Mersenne numbers. One way to find perfect numbers is to find the simpler Mersenne primes. This thesis presents the mathematical foundations necessary for understanding perfect numbers. This is followed by the definition and properties of Mersenne numbers, and the Lucas--Lehmer test for primality is mentioned and described. At the end, perfect numbers and their properties are discussed as well as multiply perfect numbers and conditions that odd perfect numbers should meet if they exist.

• Keywords: savršeni brojevi; Mersenneovi brojevi; prosti brojevi; Lucas--Lehmerov test provjere prostosti
• Keywords (english): perfect numbers; Mersenne numbers; prime numbers; Lucas--Lehmer test for primality
• Language: croatian
• URN:NBN: urn:nbn:hr:217:905897
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar edukacije matematike (sveučilišni magistar edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-02-05 12:40:52