20.500.12556/DKUM-62598 Diofantska enačba mX^2 - nY^2 = + - 1 Diophantine equation mX^2 - nY^2 = + - 1 V prvem poglavju diplomskega dela zajamemo osnovne teorije verižnih ulomkov. Posebej opišemo končne, neskončne in periodične verižne ulomke. V drugem poglavju diplomskega dela obravnavamo Pellovo enačbo oz. diofantsko enačbo oblike X^2 - dY^2 = N, kjer je N = 1 in d tako naravno število, ki ni popoln kvadrat. Osrednji del je namenjen obravnavi diofantske enačbe oblike mX^2 - nY^2 = + - 1, kjer opišemo vse pozitivne rešitve te diofantske enačbe in ugotovimo, da so slednje povezane z rešitvami Pellove enačbe R^2 - mnS^2 = 1. Ena od glavnih ugotovitev pravi, da je diofantska enačba mX^2 - nY^2 = + - 1 rešljiva natanko tedaj, ko je fundamentalna rešitev Pellove enačbe R^2 - mnS^2 = 1 kvadrat neke rešitve (x,y) enačbe mX^2 - nY^2 = + - 1. V zaključnem delu podrobno obravnavamo tiste rešitve (x_i, y_i) dotične diofantske enačbe, za katere velja, da vsi prafaktorji števila y_i delijo tudi število n. In the first chapter of the graduation thesis the basic of the theory of continued fractions are presented. Finite, infinite and periodic continued fractions are described separately. In the second chapter of the graduation thesis we consider Pell's equation, and more generally we study Diophantine equation of the form X^2 - dY^2 = N, where N = 1 and d is a positive integer that is not a perfect square. The main part of the thesis is dedicated to the examination of the Diophantine equation of the form mX^2 - nY^2 = + - 1. All positive solutions of this Diophantine equation have been described and it has been found that they are connected to the solutions of Pell’s equation R^2 - mnS^2 = 1. One of the main findings is that the Diophantine equation mX^2 - nY^2 = + - 1 is solvable if and only if the fundamental solution of Pell's equation R^2 - mnS^2 = 1 is a square of a solution (x,y) of the equation mX^2 - nY^2 = + - 1. In the concluding part those solutions (x_i, y_i) of the Diophantine equation are examined for which it is true that all the prime factors of the number y_i also divide the number n. verižni ulomek končni navadni verižni ulomek neskončni navadni verižni ulomek periodični verižni ulomek Pellova enačba diofantska enačba continued fractions finite continued fractions infinite continued fractions periodic continued fractions Pell's equations Diophantine equation. true true false [M. Vizjak] Slovenski jezik Angleški jezik Diplomsko delo 2016-08-29 22:23:42 2016-11-15 16:17:13 2022-06-29 03:09:16 2019-07-11 13:01:02 2016 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 511.512(043.2) 22764040 URN:SI:UM:DK:JCKJ8IGW UN_Vizjak_Mateja_2016.pdf UN_Vizjak_Mateja_2016.pdf 1 3C43535BF6F32B877DB834AD6C3653A3 6bb75e09acc26197a8097172d091ed16059522afc84e275b3cc98d8e1480915f 2f5bd022-7c0d-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=100901 Fakulteta za naravoslovje in matematiko 0 0 0