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master's thesis
Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija
2023. urn:nbn:hr:126:377446
Glavačević, Ana
Josip Juraj Strossmayer University of Osijek
School of Applied Mathematics and Informatics

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Glavačević, A. (2023). Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:377446

Glavačević, Ana. "Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija." Master's thesis, Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics, 2023. https://urn.nsk.hr/urn:nbn:hr:126:377446

Glavačević, Ana. "Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija." Master's thesis, Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics, 2023. https://urn.nsk.hr/urn:nbn:hr:126:377446

Glavačević, A. (2023). 'Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija', Master's thesis, Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics, accessed 10 November 2024, https://urn.nsk.hr/urn:nbn:hr:126:377446

Glavačević A. Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics; 2023 [cited 2024 November 10] Available at: https://urn.nsk.hr/urn:nbn:hr:126:377446

A. Glavačević, "Prikaz prirodnih brojeva kao suma nenegativnih k-tih potencija", Master's thesis, Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics, Osijek, 2023. Available at: https://urn.nsk.hr/urn:nbn:hr:126:377446

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Metadata
TitlePrikaz prirodnih brojeva kao suma nenegativnih k-tih potencija
Title (english)Representation Of Natural Numbers As Sums Of Non-negative K-th Powers
AuthorAna Glavačević
MentorIvan Matić (mentor)
Committee memberIvan Soldo (predsjednik povjerenstva)
Committee memberIvan Matić (član povjerenstva)
Committee memberDragana Jankov Maširević (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
School of Applied Mathematics and Informatics
Osijek
Defense date and country2023-10-18, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Other Mathematical Disciplines
Abstract
Na početku diplomskog rada govorimo o povijesnom razvoju zapisa prirodnih brojeva u obliku sume nenegativnih k-tih potencija. Mnogi matematičari proučavali su različite zapise prirodnih brojeva, a neki rezultati njihovog rada nisu ni danas dokazani. U diplomskom radu naveli smo da prost broj oblika 4k + 1 ima prikaz u obliku sume kvadrata dva cijela broja. Također, iskazali smo teorem koji govori da se jedino prirodni brojevi koji u faktorizaciji sadrže proste faktore oblika 4k + 3, k ∈ Z, s parnim eksponentom mogu zapisati kao suma kvadrata dva cijela broja. Osim toga, izveli smo i formulu za određivanje broja takvih prikaza i objasnili geometrijsku interpretaciju. Naveli smo i teorem koji govori o karakterizaciji prirodnih brojeva koji se mogu zapisati kao suma kvadrata dva prirodna broja. U četvrtom poglavlju naveden je i dokazan Gaussov teorem vezan za prirodne brojeve koji se mogu zapisati kao suma kvadrata tri cijela broja. U petom dijelu kroz Lagrangeov teorem govorimo o prikazu prirodnih brojeva u obliku sume kvadrata četiri cijela broja. U zadnjem dijelu proučavamo prikaz prirodnog broja u obliku sume kvadrata pet ili više prirodnih brojeva i navodimo rezultate koji govore o brojevima koji se mogu zapisati u tome obliku.
Abstract (english)
At the beginning of the thesis, we talk about the historical development of the notation of natural numbers in the form of the sum of non-negative k powers. Many mathematicians studied different notations of natural numbers, and some results of their work have not been proven even today. In the thesis, we stated that a prime number of the form 4k + 1 has a representation in the form of the sum of the squares of two integers. Also, we have stated a theorem that says that only natural numbers that in the factorization contain simple factors of the form 4k + 3, k ∈ Z, with an even exponent can be written as the sum of the squares of two integers. In addition, we derived the formula for determining the number of such representations and explained the geometric interpretation. We also stated a theorem that describes the characterization of natural numbers that can be written as the sum of the squares of two natural numbers. In the fourth chapter, the Gauss’s theorem related to natural numbers that can be written as the sum of the squares of three integers is stated and proved. In the fifth part, through Lagrange’s theorem, we talk about the representation of natural numbers in the form of the sum of the squares of four whole numbers. In the last part, we study the representation of a natural number in the form of the sum of the squares of five or more natural numbers and list the results that speak about the numbers that can be written in this form.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:377446
Study programmeTitle: Mathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Course: Financial and Statistical 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 on2023-11-06 12:36:05