Verižni ulomki in vprašanje Kaplanskega

diplomsko delo

Petra Žnidarič (Author), Daniel Eremita (Mentor)

Abstract

V uvodnem delu diplomskega dela je predstavljena teorija navadnih verižnih ulomkov. V nadaljevanju obravnavamo Pellove enačbe oziroma diofantske enačbe oblike x%- dy% = N, kjer sta d in N celi števili, in d tako naravno število, ki ni popolni kvadrat. Glavni del diplomskega dela je namenjen vprašanju Kaplanskega. Za praštevila p, ki jih lahko zapišemo kot vsoto popolnih kvadratov p = a%+ (2b)%, kjer sta a, b % Z, se je Kaplansky vprašal, ali sta števili a in 4b v zalogi vrednosti binarne kvadratne forme F(x,y) = x% - py%. Z drugimi besedami, ali obstajajo celo številske rešitve enačb x% - py% = a in x% - py% = 4b. Če je p praštevilo in je p % 1 (mod 4), potem se izkaže, da obstajata taka a, b % Z, da velja p = a% + (2b)%. Feit in Mollin sta dokazala, da sta števili a in 4b v zalogi vrednosti binarne kvadratne forme F(x,y) z uporabo teorije idealov. Predstavili bomo Walshevo posplošitev Feitovega izreka, ki jo je izpeljal zgolj z uporabo elementarnih metod. Kot zadnje bomo opisali še posplošitev Robertsona in Matthewsa.

Keywords

matematika;verižni ulomki;Evklidov algoritem;navadni ulomki;neskončni ulomki;periodični ulomki;vprašanje Kaplanskega;Pellova enačba;diofantska enačba;praštevila;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2011
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[P. Žnidarič]
UDC:	51(043.2)
COBISS:	19232008
Views:	1599
Downloads:	80
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	CONTINUED FRACTIONS AND A QUESTION OF KAPLANSKY
Secondary abstract:	In the first part of the graduation thesis the theory of simple continued fractions is presented. Next, we consider Pell's equations, and more generally we study Diophantine equation of the form x% - dy% = N, where N and d are integers and d is a positive integer that is not a perfect square. The main part of the thesis is devoted to a question of Kaplansky. For primes p that can be written as the sum of integer squares, p = a% + (2b)%, where a, b % Z, Kaplansky asked whether the binary quadratic form F = (x,y) = x% - py% always represents two numbers a and 4b. In other words, whether there are integer solutions to x% - py% = a and x% - py% = 4b. If p is a prime and p % 1 (mod 4), then it turns out, that there are a, b % Z, such that p = a% + (2b)%. Feit and Mollin proved that F(x,y) does always represent the numbers a and 4b using the theory of ideals. We present Walsh's generalization of the result of Feit, which was proved using only elementary methods. Finally, we describe a generalization of Robertson and Matthews.
Secondary keywords:	Euclidean algorithm;continued fractions;finite simple continued fractions;infinite simple continued fractions;periodic continued fractions;Pell's equation;a question of Kaplansky;Diophantine equation;prime number.;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	49 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	20037