20.500.12556/DKUM-10463 Diofantske četverice Diophantine quadruples 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. 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}. diofantska množica diofantska četverica regularna diofantska četverica Pellova enačba Fibonaccijevo zaporedje Diophantine set Diophantine quadruple regular Diophantine quadruple Pell`s equation Fibonacci’s sequence true true false [J. Špec] Slovenski jezik Angleški jezik Diplomsko delo 2009-05-17 11:43:24 2009-06-04 12:38:20 2022-04-11 16:18:08 2022-04-16 03:28:23 2009 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 16900872 URN:SI:UM:DK:VW70T5YD UNI_Spec_Jozica_2009.pdf UNI_Spec_Jozica_2009.pdf 1 06AF56294D132D9549D0BE45595989A2 42e2c717071f8d6f8901705b57f8864201524d444bebe1901dcc23503b84cabd d5bd5835-7c02-11eb-bb7a-00155d0001ca 20.500.12556/dkum/46dd4479-0d19-4f3b-901c-80174b8e4db3 https://dk.um.si/Dokument.php?lang=slv&id=8559 Fakulteta za naravoslovje in matematiko 0 0 0