Faktorizacija naravnih števil z binarnimi kvadratnimi formami

Kušek, Alen

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Title:	Faktorizacija naravnih števil z binarnimi kvadratnimi formami
Authors:	ID Kušek, Alen (Author) ID Eremita, Daniel (Mentor) More about this mentor...
Files:	MAG_Kusek_Alen_2018.pdf (380,44 KB) MD5: B7199F52AF3A83CBE39E13C7E1A3A3D7 PID: 20.500.12556/dkum/4036fb03-23ef-4e73-9e38-314ea3c9f71e
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
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. Predstavljen 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:	Elementarna teorija števil, faktorizacija naravnih števil, Eulerjeva formula, kvadratna forma
Place of publishing:	Maribor
Publisher:	[A. Kušek]
Year of publishing:	2018
PID:	20.500.12556/DKUM-71136
UDC:	511.17(043.2)
COBISS.SI-ID:	24020232
NUK URN:	URN:SI:UM:DK:VXAAE5OY
Publication date in DKUM:	21.09.2018
Views:	1628
Downloads:	117
Metadata:
Categories:	FNM
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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	20.07.2018

Secondary language

Language:	English
Title:	Factoring integers using binary quadratic forms
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.
Keywords:	Elemntary number theory, factoring of integers, Euler's formula, quadratic form

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