VERIŽNI ULOMKI IN VPRAŠANJE KAPLANSKEGA

Žnidarič, Petra

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Title:	VERIŽNI ULOMKI IN VPRAŠANJE KAPLANSKEGA
Authors:	ID Žnidarič, Petra (Author) ID Eremita, Daniel (Mentor) More about this mentor...
Files:	UNI_Znidaric_Petra_2011.pdf (440,43 KB) MD5: F3E14B43B87A4101C3E4C61552CA469D PID: 20.500.12556/dkum/5df0aed0-ed70-4244-bfb4-34e6e8ee4333
Language:	Slovenian
Work type:	Undergraduate thesis
Typology:	2.11 - Undergraduate Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V uvodnem delu diplomskega dela je predstavljena teorija navadnih verižnih ulomkov. V nadaljevanju obravnavamo Pellove enačbe oziroma diofantske enačbe oblike x²- dy² = 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 = a²+ (2b)², kjer sta a, b ∈ Z, se je Kaplansky vprašal, ali sta števili a in 4b v zalogi vrednosti binarne kvadratne forme F(x,y) = x² - py². Z drugimi besedami, ali obstajajo celo številske rešitve enačb x² - py² = a in x² - py² = 4b. Če je p praštevilo in je p ≡ 1 (mod 4), potem se izkaže, da obstajata taka a, b ∈ Z, da velja p = a² + (2b)². 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:	Evklidov algoritem, verižni ulomek, končni navadni verižni ulomek, neskončni navadni verižni ulomek, periodični verižni ulomek, Pellova enačba, vprašanje Kaplanskega, diofantska enačba, praštevilo.
Place of publishing:	Maribor
Publisher:	[P. Žnidarič]
Year of publishing:	2011
PID:	20.500.12556/DKUM-21284
UDC:	51(043.2)
COBISS.SI-ID:	19232008
NUK URN:	URN:SI:UM:DK:OLB0PNZU
Publication date in DKUM:	09.07.2012
Views:	2203
Downloads:	146
Metadata:
Categories:	FNM
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Secondary language

Language:	English
Title:	CONTINUED FRACTIONS AND A QUESTION OF KAPLANSKY
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 x² - dy² = 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 = a² + (2b)², where a, b ∈ Z, Kaplansky asked whether the binary quadratic form F = (x,y) = x² - py² always represents two numbers a and 4b. In other words, whether there are integer solutions to x² - py² = a and x² - py² = 4b. If p is a prime and p ≡ 1 (mod 4), then it turns out, that there are a, b ∈ Z, such that p = a² + (2b)². 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.
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.

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