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master's thesis
Kongruencije za binomne koeficijente
Zagreb: University of Zagreb, Faculty of Science, 2021. urn:nbn:hr:217:262942
Grgić, Monika
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Grgić, M. (2021). Kongruencije za binomne koeficijente (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:262942

Grgić, Monika. "Kongruencije za binomne koeficijente." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:262942

Grgić, Monika. "Kongruencije za binomne koeficijente." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:262942

Grgić, M. (2021). 'Kongruencije za binomne koeficijente', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:262942

Grgić M. Kongruencije za binomne koeficijente [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2021 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:262942

M. Grgić, "Kongruencije za binomne koeficijente", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2021. Available at: https://urn.nsk.hr/urn:nbn:hr:217:262942

Metadata
TitleKongruencije za binomne koeficijente
Title (english)Congruences for binomial coefficients
AuthorMonika Grgić
MentorTomislav Pejković (mentor)
Committee memberTomislav Pejković (predsjednik povjerenstva)
Committee memberSanja Varošanec (član povjerenstva)
Committee memberVedran Čačić (član povjerenstva)
Committee memberRobert Manger (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2021-07-21, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Primjena modularne aritmetike na binomne koeficijente interesirala je mnoge matematičare 19. stoljeća, ali i matematičare novijeg razdoblja. U radu su izloženi rezultati vezani uz ostatke pri dijeljenju binomnih koeficijenata s potencijama prostih brojeva te njihovi prikazi preko Pascalovog trokuta. U prvom poglavlju dani su osnovni pojmovi i tvrdnje potrebne za razumijevanje daljnjeg rada. U drugom poglavlju navedene su i dokazane tvrdnje matematičara Lucasa, Legendrea i Kummera. Tako su uključeni rezultati poput Lucasovog teorema koji daje jednostavnu metodu izračunavanja vrijednosti binomnog koeficijenta \(\left( \frac{m}{n}\right)\) modulo prost broj \(p\) pomoću znamenaka zapisa brojeva \(m\) i \(n\) u bazi \(p\) . U iduća dva poglavlja su iskazani i dokazani rezultati vezani uz spomenuti problem kongruencija za binomne koeficijente. U trećem poglavlju navedeni su rezultati koji se odnose na konkretne module, manje prirodne brojeve, a u četvrtom poglavlju rezultati vezani uz općenite module koji su prosti brojevi. Posljednje poglavlje posvećeno je fraktalnoj strukturi Pascalovog trokuta promatranog modulo zadani broj.
Abstract (english)
Numerous mathematicians of the nineteenth century considered problems involving binomial coeffcients in terms of modular arithmetic. In this master thesis results related to binomial coeffcients modulo a prime power as well as their representation on Pascal’s triangle are given. Collected in the first chapter are basic notions and statements necessary to understand the rest of the thesis. In the second chapter are stated and proved Lucas’, Legendre’s and Kummer’s theorem. Lucas’ theorem gives a simple method of computing binominal coeffcient \(\left( \frac{m}{n}\right)\) modulo a prime number \(p\) using the digits of base \(p\) representations of integers \(m\) and \(n\) . In the next two chapters results related to congruences for binomial coeffcients are stated and proved. In the third chapter results considering specific, smaller natural numbers as moduli are presented, while in the fourth chapter results valid for general prime moduli are given. The last chapter is devoted to the fractal structure of Pascal’s triangle modulo a fixed integer.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:262942
Study programmeTitle: 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 resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-09-15 09:39:26