Primjena linearnih formi u logaritmima na rješavanje diofantskih jednadžbi

master's thesis

• Author: Marina Ilić
• Mentor: Alan Filipin (mentor)
• Committee member: Alan Filipin (predsjednik povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-11-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu, bavimo se primjenom linearnih formi u logaritmima na rješavanje diofantskih problema. Na početku rada iskazujemo osnovne pojmove i definicije koje su nam potrebne u nastavku rada. U drugom poglavlju obrađujemo rezultate iz diofantskih aproksimacija koji se koriste u rješavanju diofantskih jednadžbi. Nadalje, u trećem poglavlju definiramo Pellovu i pellovsku jednadžbu. Također, dokazujemo niz teorema o egzistenciji i strukturi rješenja navedenih jednadžbi. U posljednjem poglavlju bavimo se linearnim formama, tj. jednom od modernih metoda rješavanja diofantskih jednadžbi. Linearna forma u logaritmima algebarskih brojeva je izraz oblika \(b_{1}\log{\alpha_{1}} +...+ b_{n}\log{\alpha_{n}}\), gdje su \(\alpha_{i}\) algebarski brojevi, a \(b_{i}\) cijeli brojevi. Prilikom rješavanja diofantskih jednadžbi želimo odrediti donju ogradu za linearnu formu u logaritmima. Pritom, za određivanje donje ograde koristimo Bakerovu teoriju. Na primjerima ćemo pokazati kako koristimo linearne forme u logaritmima za rješavanje simultanih pellovskih jednadžbi te kako ih primjenjujemo na Fibonaccijeve brojeve.

Abstract (english)

In this graduate thesis, we are dealing with linear forms in logarithms to solve Diophantine problems. At the beginning of this thesis we present the basic concepts and definitions we need to continue the work. In the second chapter, we discuss the results of Diophantic approximations used in solving Diophantine equations. Furthermore, in the third chapter we define the Pell’s and generalized Pell equations. We also prove a series of theorems about existence and structure of the solutions of the equations mentioned. In the last chapter, we are dealing with linear forms, one of the modern methods of solving Diophantine equations. Linear form in logarithms of algebraic numbers is a form expression \(b_{1}\log{\alpha_{1}} +...+ b_{n}\log{\alpha_{n}}\), where \(\alpha_{i}\) are algebraic numbers, and \(b_{i}\) integers. When solving Diophantine equations, we want to determine the lower fence for linear form in logarithms. In doing so, we use Baker’s theory to determine the bottom fence. In the examples we will show how we use linear forms in logarithms to solve simultaneous generalized Pell equations and how we apply them to Fibonacci numbers.

• Keywords: linearne forme; diofantski problemi; diofantske jednadžbe; Pelleova jednadžba; pellovska jednadžba; Bakerova teorija; Fibonaccijevi brojevi
• Keywords (english): linear forms; Diophantine problems; Diophantine equations; Pell’s equations; generalized Pell’s equations; Baker’s theory; Fibonacci numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:444909
• 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: 2019-05-09 11:35:40