Faktorizacija naravnih števil z binarnimi kvadratnimi formami

magistrsko delo

Alen Kušek (Author), Daniel Eremita (Mentor)

Abstract

V magistrskem delu obravnavmo faktorizacijo naravnih števil oblike ▫$mx^2 + ny^2$▫. Delo je razdeljeno na štiri poglavja. V prvem poglavju spoznamo Fermatovo faktorizacijsko metodo in Gaussova cela števila. V drugem poglavju se ukvarjamo s faktorizacijo števil oblike ▫$mx^2 + ny^2$▫. Predstavljena je Eulerjeva formula, s katero je mogoče faktorizirati števila oblike ▫$mx^2 + ny^2$▫. Prav tako obravnavamo sodobnejšo metodo faktorizacije, ki sta jo razvila Lucas in Mathews. Pred-stavljen je enostaven dokaz njunega izreka, ki ga je podal Brillhart. V tretjem poglavju raziskujemo faktorizacijo lihega naravnega števila, ki ga lahko zapišemo s kvadratno formo ▫$mx^2 + ny^2$▫ na dva različna načina, kjer sta ▫$m$▫ in ▫$n$▫ naravni števili. Pri tem podamo eksplicitno formulo za faktorizacijo in pogoje za kvadratno formo, ki so potrebni za obstoj te formule. Pri tem bomo uporabljali rezultate prejšnjega poglavja. V zadnjem poglavju obravnavamo podoben problem kot v tretjem poglavju, le da tokrat predpostavimo, da je ▫$n$▫ negativno celo število.

Keywords

magistrska dela;elementarna teorija števil;faktorizacija naravnih števil;Eulerjeva formula;kvadratna forma;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[A. Kušek]
UDC:	511.17(043.2)
COBISS:	24020232
Views:	672
Downloads:	81
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Factoring integers using binary quadratic forms
Secondary abstract:	In the master's thesis we consider the factoring of positive integers of the form ▫$mx^2 + ny^2$▫. The work is divided into four chapters. In the first section, we learn about Fermat's factorization method and Gaussian integers. In the second chapter, we are dealing with the factoring of positive integers of the form ▫$mx^2 + ny^2$▫. The Euler formula is presented, which enables the factorization of integers of the form ▫$mx^2 + ny^2$▫. We are also considering a more modern factorization method, developed by Lucas and Mathews. A simple proof of their theorem given by Brillhart is also presented. In the third chapter, we consider the factorization of an odd integer N that has a double representation by quadratic forms ▫$mx^2 + ny^2$▫, where ▫$m$▫ and ▫$n$▫ are natural numbers. We give an explicit formula for the factorization and the quadratic form conditions necessary for the existence of this formula. We will use the results of the previous chapter. In the final chapter we deal with a similar problem as in Chapter 3, but this time we suppose that ▫$n$▫, is a negative integer.
Secondary keywords:	master theses;elementary number theory;factoring of integers;Euler's formula;quadratic formula;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	VIII, 51 f.
ID:	10949101