Porazdelitev kvadratnih ostankov : magistrsko delo

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Title:	Porazdelitev kvadratnih ostankov : magistrsko delo
Authors:	ID Možina, Maja (Author) ID Eremita, Daniel (Mentor) More about this mentor...
Files:	EMAG_Mozina_Maja_2024.pdf (1,51 MB) MD5: 98215F7946F7D5F72F78C57114593FC9
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Kvadratni ostanek lihega praštevila p je tako celo število a, da ima kongruenčna enačba x^2≡a (mod p) vsaj eno rešitev x, pri čemer sta si števili a in p tuji. Če omenjena enačba nima rešitve, je a kvadratni neostanek lihega praštevila p. V magistrskem delu proučujemo kvadratne ostanke lihega praštevila p ter njihove lastnosti. V prvem delu spoznamo Legendrov simbol in njegove lastnosti ter Eulerjev kriterij za določanje vrednosti Legendrovega simbola. Nato se vprašamo, kdaj je število -1 kvadratni ostanek lihega praštevila p in kdaj je število 2 kvadratni ostanek lihega praštevila p. Kasneje spoznamo še modularne kvadratne korene, torej med seboj nekongruentne rešitve enačb oblike x^2≡a (mod pq), kjer sta p in q različni lihi praštevili. To tehniko prikažemo v praktičnem primeru – elektronski met kovanca. V drugem delu se posvetimo iskanju parov in trojic zaporednih naravnih števil, ki so kvadratni ostanki lihega praštevila p. Prav tako obravnavamo pare zaporednih naravnih števil, ki so kvadratni neostanki lihega praštevila p, ter takšne pare zaporednih naravnih števil, kjer je eno kvadratni ostanek, drugo pa kvadratni neostanek lihega praštevila p.
Keywords:	Praštevila, kvadratni ostanki, kvadratni neostanki, Legendrov simbol, kongruenčna enačba, pari zaporednih kvadratnih ostankov, trojice zaporednih kvadratnih ostankov, modularni kvadratni koren.
Place of publishing:	Maribor
Place of performance:	Maribor
Publisher:	M. Možina
Year of publishing:	2024
Number of pages:	38 f.
PID:	20.500.12556/DKUM-87171
UDC:	511.17(043.2)
COBISS.SI-ID:	189019907
Publication date in DKUM:	18.03.2024
Views:	330
Downloads:	61
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:	28.02.2024

Secondary language

Language:	English
Title:	Distribution of quadratic residues : na enovitem magistrskem študijskem programu Predmetni učitelj, usmeritev izobraževalna matematika
Abstract:	Quadratic residue of an odd prime p is a positive integer a such that the congruence x^2≡a (mod p) has at least one solution x, where integers a and p are relatively prime. If the congruence has no such solution, then the integer a is called a quadratic nonresidue modulo p. In this master thesis we study quadratic residues of odd primes p and some of their properties. In the first part we focus on the Legendre symbol and its' properties as well as the Euler criterion that helps us determine the value of Legendre symbols. Then we pose a question; when is the integer -1 a quadratic residue modulo p and when is the integer 2 a quadratic residue modulo p. After that we move on to modular square roots, the four noncongruent solutions of congruences x^2≡a (mod pq), where p and q represent two different odd primes. This technique is then used in a practical example – flipping coins electronically. In the second part of the thesis we focus on finding two or three consecutive positive integers that are quadratic residues modulo p. We also consider pairs of consecutive positive integers where both are quadratic nonresidues modulo p and such pairs where one of the integers is a residue modulo p and the other is not.
Keywords:	quadratic resiues, quadratic nonresidues, prime numbers, congruence, Legendre symbol, pairs of consecutive quadratic residues, triplets of consecutive quadratic residues, modular square roots.

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