Diofantske četverice

diplomsko delo

Jožica Špec (Author), Daniel Eremita (Mentor)

Abstract

Diofantska množica S je množica takih naravnih števil, da je x * y + 1 popolni kvadrat, za vse x različne od y iz množice S. Diofantski množici s štirimi elementi pravimo diofantska četverica. Problem diofantskih četveric je v tretjem stoletju prvi predstavil grški matematik Diofant iz Aleksandrije. Namen diplomskega dela je opisati vse regularne diofantske četverice oblike {1, b, c, d}, kjer je 1 < b < c < d, ter izpeljati algoritme za njihovo konstrukcijo. Prvo poglavje je namenjeno reševanju Pellovih enačb, saj moramo za konstrukcijo vseh regularnih diofantskih četveric, oblike {1, b, c, d}, najprej rešiti nekaj Pellovih enačb oblike x^2-d^2=L, kjer je L različen od +1 ali -1, katere imajo več neskončnih družin rešitev. V drugem poglavju je predstavljen problem diofantskih četveric. Poglavje opisuje zgodovinsko ozadje raziskovanja na problemu diofantskih četveric. Opisana je povezava med Fibonaccijevim zaporedjem in diofantskimi četvericami. Predstavljen je problem nadgradnje diofantske trojke do diofantske četverice. V tretjem poglavju je predstavljena konstrukcija neskončne družine regularnih diofantskih četveric oblike {1, b, c, d}, kjer je 1 < b < c < d. V četrtem poglavju karakteriziramo vse regularne diofantske četverice oblike {1, b, c, d} in podamo dva algoritma za njihovo konstrukcijo.

Keywords

matematika;diofantske množice;diofantska četverica;Pellova enačba;Fibonaccijeva zaporedja;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2009
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[J. Špec]
UDC:	51(043.2)
COBISS:	16900872
Views:	2391
Downloads:	165
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Diophantine quadruples
Secondary abstract:	A set S of positive integers is said to have a Diophantine property, and called a Diophantine set, if x * y + 1 is a perfect square for any x different from y, where x and y belongs to the set S. Diophantine set with four elements is called Diophantine quadruple. The problem of Diophantine quadruples was originally posed by the Greek mathematician Diophantus from Aleksandria in the third century. Our purpose is to describe all regular Diophantine quadruples of the form {1, b, c, d}, where 1 < b < c < d, and also to obtain algorithms for their construction. First chapter describes the subject of Pell's equations. In order to generate all regular Diophantine quadruples emanating from 1, i. e., {1, b, c, d}, we need to solve some non-unit Pell equations which have several infinitive families of solution. In the second chapter the problem of Diophantine quadruples is presented. The chapter describes historical background research about the problem of Diophantine quadruples. It includes explanation of the connection between Fibonacci's sequence and Diophantine quadruples. The problem of upgrading a Diophantine triple to Diophantine quadruple is also considered. In the third chapter, the construction of infinite family of regular Diophantine quadruples of the form {1, b, c, d}, where 1 < b < c < d, is presented. In the fourth chapter we characterize the regular Diophantine quadruples emanating from 1, and the chapter includes the description of two algorithms for construction of regular Diophantine quadruples of the form {1, b, c, d}.
Secondary keywords:	Diophantine set;Diophantine quadruple;regular Diophantine quadruple;Pell`s equation;Fibonacci’s sequence;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	VII, 40 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	17821