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master's thesis
Prosti brojevi u aritmetičkim nizovima
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:566452
Škoro, Igor
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Škoro, I. (2022). Prosti brojevi u aritmetičkim nizovima (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:566452

Škoro, Igor. "Prosti brojevi u aritmetičkim nizovima." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:566452

Škoro, Igor. "Prosti brojevi u aritmetičkim nizovima." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:566452

Škoro, I. (2022). 'Prosti brojevi u aritmetičkim nizovima', Master's thesis, University of Zagreb, Faculty of Science, accessed 26 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:566452

Škoro I. Prosti brojevi u aritmetičkim nizovima [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2024 December 26] Available at: https://urn.nsk.hr/urn:nbn:hr:217:566452

I. Škoro, "Prosti brojevi u aritmetičkim nizovima", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:566452

Metadata
TitleProsti brojevi u aritmetičkim nizovima
Title (english)Prime numbers in arithmetic sequences
AuthorIgor Škoro
MentorMarko Tadić (mentor)
Committee memberMarko Tadić (predsjednik povjerenstva)
Committee memberAleksandra Čižmešija (član povjerenstva)
Committee memberLjiljana Arambašić (član povjerenstva)
Committee memberIgor Pažanin (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-10-12, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U radu se proučavaju prosti brojevi u aritmetičkim nizovima. Glavni cilj rada je pokazati koji artimetički nizovi sadrže beskonačno mnogo prostih brojeva. U prvom poglavlju, navesti ćemo osnovne definicije i dokaze iz polja algebre, teorije brojeva i kompleksne analize, koji će nam trebati u ostatku rada. U drugom poglavlju cilj nam je dokazati Dirichletov teorem o prostim brojevima, koji govori da aritmetički niz ak+m, k ∈ \(\mathbb{N}\) sadrži beskonačno mnogo prostih brojeva ukoliko su a i m relativno prosti. U ovom poglavlju, prvo ćemo razmotriti nekoliko aritmetičkih nizova, za koje je jednostavno pokazati da sadrže beskonačno mnogo prostih brojeva. Zatim ćemo uvesti pojam Karaktera konačne Abelove grupe, Dirichletovih redova, L-funkcija i Dirichletove gustoće, koji će nam biti potrebni da onda dokažemo Dirichletov teorem. Na kraju, navesti ćemo primjenu Dirichletovog teorema u teoriji brojeva.
Abstract (english)
In this thesis we examine prime numbers in arithmetic progressions. The main purpose of the thesis is to show which arithmetic progressions contain infinitely many prime numbers. In the first chapter, we will state definitions and theorems from the fields of algebra, number theory and complex analysis, which we will use in the rest of the thesis. In the second chapter, our goal is to prove Dirichlet’s theorem on arithmetic progression, which states that the arithmetic progression ak + m, k ∈ \(\mathbb{N}\), contains infinitely many primes if a and m are relatively prime. In this chapter, we will first consider a few arithmetic progressions, for which it is easy to show that they contain infinitely many primes. Next, we will introduce the ideas of group characters, Dirichlet series, L-functions and Dirichlet density, which we will then use to prove Dirichlet’s theorem. Finally, we will show an application of Dirichlet’s theorem in number theory.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:566452
Study programmeTitle: Pure Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2022-11-09 12:12:19