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Title:Diofantske četverice
Authors:ID Špec, Jožica (Author)
ID Eremita, Daniel (Mentor) More about this mentor... New window
Files:.pdf UNI_Spec_Jozica_2009.pdf (347,41 KB)
MD5: 06AF56294D132D9549D0BE45595989A2
PID: 20.500.12556/dkum/46dd4479-0d19-4f3b-901c-80174b8e4db3
 
Language:Slovenian
Work type:Undergraduate thesis
Organization:FNM - Faculty of Natural Sciences and Mathematics
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:diofantska množica, diofantska četverica, regularna diofantska četverica, Pellova enačba, Fibonaccijevo zaporedje
Place of publishing:Maribor
Publisher:[J. Špec]
Year of publishing:2009
PID:20.500.12556/DKUM-10463 New window
UDC:51(043.2)
COBISS.SI-ID:16900872 New window
NUK URN:URN:SI:UM:DK:VW70T5YD
Publication date in DKUM:04.06.2009
Views:3091
Downloads:193
Metadata:XML DC-XML DC-RDF
Categories:FNM
:
ŠPEC, Jožica, 2009, Diofantske četverice [online]. Bachelor’s thesis. Maribor : J. Špec. [Accessed 23 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=10463
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Secondary language

Language:English
Title:Diophantine quadruples
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}.
Keywords:Diophantine set, Diophantine quadruple, regular Diophantine quadruple, Pell`s equation, Fibonacci’s sequence


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