20.500.12556/RUL-159247 Sylvestrovo zaporedje magistrsko delo Sylvester's Sequence V tem magistrskem delu predstavimo Sylvestrovo zaporedje. Sylvestrovo zaporedje je zaporedje naravnih števil, v katerem je vsak naslednji člen produkt predhodnih členov, povečan za ena. Osredotočimo se na njegove lastnosti in jih tudi dokažemo. Predstavimo še nerešen problem v povezavi s Sylvestrovim zaporedjem. V nadaljevanju obravnavamo zaporedje obratnih vrednosti členov Sylvestrovega zaporedja, ki tvorijo številsko vrsto. Raziščemo povezavo med to številsko vrsto in egipčanskimi ulomki. Predstavimo tudi diofantsko enačbo, katere rešitve so navzgor omejene s členi Sylvestrovega zaporedja, zmanjšanimi za 1, ki jo povežemo še z dvema matematičnima problemoma. V zadnjem poglavju obravnavane teme združimo v zbirko nalog, ki jih učitelji matematike lahko uporabijo pri pripravah na matematično tekmovanje za Vegovo priznanje za srednje šole ali pri dodatnem pouku matematike v srednji šoli. In this master's thesis, we present Sylvester's sequence. Sylvester's sequence is a sequence of natural numbers in which each subsequent term is the product of all previous terms, increased by one. We focus on properties of this sequence and provide their proofs. We also present an unsolved problem related to the Sylvester's sequence. Furthermore, we examine the sequence of the reciprocals of the terms of the Sylvester's sequence, which form a series. We explore the connection between the terms of this series and Egyptian fractions. We also present a diophantine equation, whose solutions are bounded above by the terms of the Sylvester's sequence, reduced by one, and relate it to two separate mathematical problems. In the final chapter, we combine the topics previously discussed into a collection of problems that teachers of mathematics can use to prepare high school students for the Vega Prize mathematics competition or for additional mathematics classes in high school. Sylvestrovo zaporedje egipčanski ulomki praštevila diofantska enačba matematika Sylvester's sequence Egyptian fractions prime numbers diophantine equation true false false T. Narat Slovenski jezik Angleški jezik Magistrsko delo/naloga 2024-07-04 08:30:29 2024-07-04 08:30:31 2024-07-05 10:51:52 0000-00-00 00:00:00 2024 0 Kamnik 0 Kamnik 59 str. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 1970-01-01 511(043.2) 162177 200829187 32583.pdf 32583.pdf 1 EE4213E7468D3EF4769A8572EC272FAB a892fec3565bc00a492af39e19ea22400cdf5bccf19e0377713ba283c9d36aaa e9092ab8-39ce-11ef-90ba-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=187204 Pedagoška fakulteta 0 0 0