Wolstenholmeov teorem

master's thesis

• Author: Lucija Jović
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Joseph Wolstenholme engleski je matematičar koji je 1862. godine dokazao da je za prost broj \(p\) veći od 3 brojnik racionalnog broja \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\) djeljiv s \(p^{2}\). Ovaj diplomski rad posvećen je Wolstenholmeovom teoremu i sadrži četiri poglavlja. U prvom poglavlju izlažemo osnovne pojmove i rezultate iz teorije brojeva koji se koriste u nastavku rada. U drugom poglavlju iskazujemo i dokazujemo Wolstenholmeov teorem te navodimo tvrdnju ekvivalentnu navedenom teoremu. Treće poglavlje posvećeno je nekim generalizacijama Wolstenholmeovog teorema. Navedeno je i dokazano nekoliko varijacija, a zatim dokazana i generalizacija. U zadnjem poglavlju ovog rada pokazujemo kako obrat Wolstenholmeovog teorema vrijedi u određenim beskonačnim skupovima prirodnih brojeva.

Abstract (english)

Joseph Wolstenholme was an English mathematician who proved in 1862 that for a prime \(p\) greater than 3, the numerator of the fraction \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\) is divisible by \(p^{2}\). This graduate thesis is devoted to the Wolstenohlme’s theorem and contains four chapters. In the first chapter, we give some of the fundamental terms and results from the theory of numbers used throughout the rest of the thesis. In the second chapter we state and prove Wolstenholme’s theorem and give an equivalent formulation. The third chapter gives some generalizations of Wolstenholme’s theorem. Several variations are stated and proved, followed by a generalization. In the last chapter of this graduate thesis, we prove that the converse of Wolstenholme’s theorem is true in certain infinite sets of natural numbers.

• Keywords: Wolstenholmeov teorem; generalizacija teorema; obrat teorema
• Keywords (english): Wolstenohlme’s theorem; generalizations of theorem; converse of theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:354084
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-07-02 12:57:26