DIOFANT-DAVENPORTOV PROBLEM ZA GAUSSOVA CELA ŠTEVILA

Bezjak, Katja

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Title:	DIOFANT-DAVENPORTOV PROBLEM ZA GAUSSOVA CELA ŠTEVILA
Authors:	ID Bezjak, Katja (Author) ID Eremita, Daniel (Mentor) More about this mentor...
Files:	UNI_Bezjak_Katja_2011.pdf (277,30 KB) MD5: 1C63FB2B07CE426D15104C079D31F6EB PID: 20.500.12556/dkum/8c1fb974-d61f-4c04-a2a9-becd3b14c503
Language:	Slovenian
Work type:	Undergraduate thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Grški matematik Diofant je ugotovil, da ima množica racionalnih števil {1/16,33/16,17/4 ,105/16} tako lastnost, da je produkt poljubnih dveh različnih števil iz te množice povečan za 1, kvadrat racionalnega števila. Prvo množico naravnih števil z zgornjo lastnostjo je našel Fermat. To je množica {1,3,8,120}. Davenport in Baker pa sta leta 1969 dokazala, da če je d tako naravno število, da ima množica {1,3,8,d} Diofantovo lastnost, potem je d =120. Naj bo z Gaussovo celo število in naj bo m≥2 naravno število. Množica neničelnih Gaussovih celih števil {a_1,a_2,..,a_m } ima lastnost D(z), če je produkt poljubnih dveh različnih elementov te množice povečan za z, kvadrat Gaussovega celega števila. Taki množici pravimo kompleksna diofantska m-terica z lastnostjo D(z). V diplomskem delu bomo dokazali (izrek 3.2.1), da v primeru, ko je b liho celo število ali če je a≡b≡2 (mod 4), potem kompleksna diofantska četvorka z lastnostjo D(a+bi) ne obstaja. Dokazali bomo tudi (posledica 3.2.2), da obstajata vsaj dve neekvivalentni kompleksni diofantski četvorki z lastnostjo D(z), če je Gaussovo celo število z možno zapisati kot razliko kvadratov dveh Gaussovih celih števil in z∉{±2,±1±2i,±4i}. Zadnji del diplomskega dela je namenjen obravnavi kompleksnih diofantskih četvork z lastnostjo D(l^2 ). Dokazali bomo tudi ( izrek 3.3.2), da vsak diofantski kompleksni par {a,b} z lastnostjo D(l^2), kjer ab ni popoln kvadrat, lahko razširimo do kompleksne diofantske četvorke z lastnostjo D(l^2) na neskončno načinov.
Keywords:	Diofant-Davenportov problem, Gaussova cela števila, diofantska četvorka, kompleksna diofantska četvorka.
Place of publishing:	Maribor
Publisher:	[K. Bezjak]
Year of publishing:	2011
PID:	20.500.12556/DKUM-18194
UDC:	51(043.2)
COBISS.SI-ID:	18507528
NUK URN:	URN:SI:UM:DK:4BEISPI3
Publication date in DKUM:	07.07.2011
Views:	2939
Downloads:	196
Metadata:
Categories:	FNM
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Secondary language

Language:	English
Title:	THE PROBLEM OF DIOPHANTUS AND DAVENPORT FOR GAUSSIAN INTEGERS
Abstract:	The Greek mathematician Diophantus noted that the set of rational numbers {1/16,33/16,17/4 ,105/16} has the following property: the product of any two distinct numbers in this set increased by 1 is a square of a rational number. Fermat first found a set of four positive integers with the above property, and it was {1,3,8,120}. Later, Davenport and Baker showed that if d is a positive integer such that the set {1,3,8,d} has the property of Diophantus, then d has to be 120. Let z be a Gaussian integer and let m≥2 be an integer. A set of nonzero Gaussian interegs {a_1,a_2,..,a_m } is said to have the property D(z) if the product of any two distinct elements increased by z is a square of a Gaussian integer. Such a set is called a complex Diophantine m-tuple with the property D(z). In this graduation thesis we prove (Theorem 3.2.1) that if b is an odd integer or a≡b≡2 (mod 4), then there does not exist a complex Diophantine quadruple with the property D(a+bi). We also prove (Theorem 3.2.2) that if a Gaussian integer z is representable as a difference of the squares of two Gaussian integer and z∉{±2,±1±2i,±4i}, then there exist at least two nonequivalent complex Diophantine quadruples with the property D(z). In the last part of the thesis we consider complex Diophantine quadruples with the property D(l^2 ). We shall see (Theorem 3.3.2) that every complex Diophantine pair {a,b} with the property D(l^2 ), where ab is not a perfect square, can be extended to the complex Diophantine quadruple with the same property in an infinite number of ways.
Keywords:	Diofant-Davenport problem, Gaussian integers, Diophantine quadruple, complex Diophantine quadruple

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