Schurova števila

diplomsko delo

Urška Lamovec (Author), Boštjan Kuzman (Mentor)

Abstract

V diplomskem delu podrobneje obravnavamo Schurov izrek o vsot-prostih particijah in definiramo n-to Schurovo število S(n) kot največje naravno število, za katerega obstaja razbitje množice {1,...,S(n)} na n disjunktnih vsot-prostih podmnožic. Zapišemo prvih nekaj znanih Schurovih števil in določimo meje, znotraj katerih se gibljejo vrednosti večjih, še neznanih Schurovih števil. Omenimo šibka Schurova števila. Schurov izrek formuliramo tudi kot problem barvanja in posledico Ramseyjeve teorije. Za konec si pogledamo, kako je Schurov izrek povezan z zadnjim Fermatovim izrekom. Pokažemo, na kakšen način je Schur poenostavil Dicksonovo trditev, da ima enakost x^n+y^n=z^n pri danem naravnem številu n > 2 netrivialne rešitve v Z_p za vsa dovolj velika praštevila p.

Keywords

kombinatorični izrek o ničlah;polinomi;kombinatorika;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[U. Lamovec]
UDC:	51(043.2)
COBISS:	12151113
Views:	601
Downloads:	100
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Schur Numbers
Secondary abstract:	In the thesis, Schur's theorem on sum-free partitions is proven and Schur number S(n) is defined as the largest positive integer with the property that the set {1,...,S(n)} can be partitioned into n sum-free subsets. Values of known Schur numbers S(1) to S(5) are given as well as some upper and lower bounds for general S(n). Weak Schur numbers are also defined. Moreover, Schur's theorem is formulated as a graph coloring problem and presented as a corollary of Ramsey theorem. In conclusion, Schur's theorem is linked to Fermat's last theorem. Schur's simplification of Dickson's proof that equation x^n+y^n=z^n for fixed n > 2 has nontrivial solutions in Z_p for all sufficiently large prime p is given.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj, Matematika in fizika
Pages:	42 str.
ID:	10973414