Jacobijev simbol

master's thesis

• Title (english): Jacobi's symbol
• Author: Magdalena Boljfetić
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Jacobijev simbol generalizacija je Legendreovog simbola \(\left( \frac{a}{p}\right)\) , gdje je \(a\) cijeli broj i \(p\) neparan prost broj, kojeg defniramo kao 1, -1 ili 0 ovisno o tome je li kvadratni ostatak modulo \(p\) , kvadratni neostatak modulo \(p\) ili \(p\) dijeli \(a\). U ovom diplomskom radu predstavljena su i dokazana brojna svojstva Jacobijevog simbola. Zajedno s kvadratnim zakonom reciprociteta ta se svojstva mogu koristiti za vrlo učinkovito računavanje vrijednosti Jacobijevog simbola. Također, Jacobijev simbol koristi se u testovima prostosti i u defniciji pseudoprostih brojeva.

Abstract (english)

The Jacobi symbol is a generalization of the Legendre symbol \(\left( \frac{a}{p}\right)\), where \(a\) is an integer and \(p\) is an odd prime, which is defned to be equal to 1, -1 or 0 depending on whether a is a quadratic residue modulo \(p\) , a quadratic nonresidue modulo \(p\) or \(p\) divides \(a\). In this master's thesis a number of useful properties of the Jacobi symbol are presented and proved. Together with the law of quadratic reciprocity, they can be used to compute the Jacobi symbol very effciently. Also, Jacobi symbols are used in primality testing and in the defnition of a type of pseudoprime.

• Keywords: Jacobijev simbol; Legendreov simbol; pseudoprosti brojevi
• Keywords (english): Jacobi symbol; Legendre symbol; pseudoprime
• Language: croatian
• URN:NBN: urn:nbn:hr:217:025267
• 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: 2021-09-14 11:25:27