diplomsko delo
Abstract
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 [na] 2 - dY [na] 2 = N, kjer je N = 1 in d tako naravno število, ki ni popoln kvadrat. Osrednji del je namenjen obravnavi diofantske enačbe oblike mX [na] 2 - nY [na] 2 =[plusminus]1, kjer opišemo vse pozitivne rešitve te diofantske enačbe in ugotovimo, da so slednje povezane z rešitvami Pellove enačbe R [na] 2 - mnS [na]2 = 1. Ena od glavnih ugotovitev pravi, da je diofantska enačba mX [na] 2 - nY [na] 2 =[plusminus]1 rešljiva natanko tedaj, ko je fundamentalna rešitev Pellove enačbe R [na] 2 - mnS [na] 2 = 1 kvadrat neke rešitve (x,y) enačbe mX [na] 2 - nY [na] 2 = 1. V zaključnem delu podrobno obravnavamo tiste rešitve (x [spodaj] i, y [spodaj] i) dotične diofantske enačbe, za katere velja, da vsi prafaktorji števila y [spodaj] i delijo tudi število n.
Keywords
verižni ulomek;končni navadni verižni ulomek;neskončni navadni verižni ulomek;periodični verižni ulomek;Pellova enačba;diofanstska enačba;diplomska dela;
Data
Language: |
Slovenian |
Year of publishing: |
2016 |
Typology: |
2.11 - Undergraduate Thesis |
Organization: |
UM FNM - Faculty of Natural Sciences and Mathematics |
Publisher: |
[M. Vizjak] |
UDC: |
511.512(043.2) |
COBISS: |
22764040
|
Views: |
1074 |
Downloads: |
76 |
Average score: |
0 (0 votes) |
Metadata: |
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Other data
Secondary language: |
English |
Secondary title: |
Diophantine equation mX^2 - nY^2 = + - 1 |
Secondary abstract: |
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 [na] 2 - dY [na] 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 [na] 2 - n[na] 2 =[plusminus]1. All positive solutions of this Diophantine equation have been described and it has been found that they are connected to the solutions of Pellʼs equation R [na] 2 - mnS [na] 2 = 1. One of the main findings is that the Diophantine equation mX [na] 2 - nY [na] 2 =[plusminus]1 is solvable if and only if the fundamental solution of Pell's equation R [na] 2 - mnS [na] 2 = 1 is a square of a solution (x,y) of the equation mX [na] 2 - nY [na] 2 =[plusminus]1. In the concluding part those solutions (x [spodaj] i, y [spodaj] i) of the Diophantine equation are examined for which it is true that all the prime factors of the number y [spodaj] i also divide the number n. |
Secondary keywords: |
continued fractions;finite continued fractions;infinite continued fractions;periodic continued fractions;Pellʼs equations;Diophantine equation;theses; |
URN: |
URN:SI:UM: |
Type (COBISS): |
Undergraduate thesis |
Thesis comment: |
Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo |
Pages: |
59 f. |
ID: |
9164988 |