Kongruencije za binomne koeficijente

master's thesis

• Title (english): Congruences for binomial coefficients
• Author: Monika Grgić
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Robert Manger (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-21, 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: modularna aritmetika; Pascalov trokut; Lucas; Legende; Kummer
• Keywords (english): modular arithmetic; Pascal’s triangle; Lucas’ theorem; Legendre’s theorem; Kummer’s theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:262942
• 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: 2021-09-15 09:39:26