Porazdelitev kvadratnih ostankov

magistrsko delo

Maja Možina (Author), Daniel Eremita (Mentor)

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 števili a in p tuji. Če omenjena enačba nima rešitve, je a kvadratni ne ostanek lihega praštevila p. V magistrskem delu proučujemo kvadratne ostanke lihega praštevila p in njihove lastnosti. V prvem delu spoznamo Legendrov simbol in njegove lastnosti in 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 tudi modularne kvadratne korene, torej med seboj ne kongruentne 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 ne ostanki lihega praštevila p, ter takšne pare zaporednih naravnih števil, kjer je eno kvadratni ostanek, drugo pa kvadratni ne ostanek lihega praštevila p.

Keywords

magistrska dela;praštevila;kvadratni ostanki;kvadratni neostanki;Legendrov simbol;kongruenčna enačba;pari zaporednih kvadratnih ostankov;trojice zaporednih kvadratnih ostankov;modularni kvadratni koren;

Data

Language:	Slovenian
Year of publishing:	2024
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Možina]
UDC:	511.17(043.2)
COBISS:	189019907
Views:	32
Downloads:	4
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Distribution of quadratic residues
Secondary 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.
Secondary keywords:	master theses;quadratic resiues;quadratic nonresidues;prime numbers;congruence;Legendre symbol;pairs of consecutive quadratic residues;triplets of consecutive quadratic residues;modular square roots;Teorija števil;Praštevila;Univerzitetna in visokošolska dela;
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	38 f.
ID:	23018770