Uvod v teorijo Gaussovih fakultet

diplomsko delo

Urška Vučak (Author), Daniel Eremita (Mentor)

Abstract

V diplomskem delu je obravnavana Gaussova fakulteta N_n!, ki je definirana kot produkt vseh naravnih števil do števila N, ki so tuja s številom n. Na začetku so predstavljeni osnovni pojmi elementarne teorije števil, ki so potrebni za razumevanje nadaljnje obravnave. V drugem poglavju obravnavamo Wilsonov izrek in Gaussovo posplošitev tega izreka ter definiramo Gaussovo fakulteto. Osrednji del diplomskega dela je tretje poglavje, v katerem posebno pozornost namenimo Gaussovi fakulteti oblike ((n-1)/M_n! in delnim produktom, ki jih dobimo, ko produkt (n-1)_n! razdelimo na M enakih delov. Najprej se omejimo na praštevila, nato opazujemo delne produkte števila (n-1)_n! in se vprašamo, kdaj so vsi med seboj kongruentni. Za konec dokažemo še dve domnevi iz začetka poglavja s pomočjo Gaussovega in Jacobijevega izreka o binomskih koeficientih in zaključimo z njunimi razširitvami.

Keywords

diplomska dela;Wilsonov izrek;Gaussov izrek;Gaussova fakulteta;Eulerjeva funkcija;praštevilo;kongruence;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[U. Vučak Markež]
UDC:	511(043.2)
COBISS:	22758152
Views:	979
Downloads:	124
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	An introduction to Gauss factorials
Secondary abstract:	In the graduation thesis we study Gauss factorial N_n!, which is defined as the product of all positive integers up to N, that are relatively prime to n. At first the basic concepts of elementary number theory necessary for understanding the following treatment are presented. In the second chapter Wilson's theorem and its generalization by Gauss are presented and Gauss factorial is defined. The main part of the graduation thesis is the third chapter in which the special treatment is on Gauss factorials ((n-1)/M)_n! and partial products which are obtained when the product (n-1)_n! is divided into M equal parts. At first we restrict ourselfs on primes, then we observe the partial products of (n-1)_n! and we ask ourselfs, when they are all congruent to each other. At the end the two assumptions from the beginning of the chapter are proved with the Gauss's and Jacobi's binomial coeficient theorem and presenting their extensions we end the chapter.
Secondary keywords:	theses;Wilson's theorem;Gauss's theorem;Gauss factorial;Euler function;prime number;congruence;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	IX, 47 f.
ID:	9164932