Balansirajući brojevi

master's thesis

• Title (english): Balancing numbers
• Author: Lucija Plantak
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu definiramo balansirajući broj \( n \in \mathbb{N}\) kao rješenje diofantske jednadžbe 1 + 2 + ... + (n + 1) = (n + 1) + (n + 2) + ... + (n + r); gdje je \( r \in \mathbb{N}_0\) balanser pridružen broju \( n \). Na primjer, 1; 6; 35 i 204 su balansirajući brojevi s balanserima 0; 2; 14 and 84, redom. Zanimljivo je da balansirajući brojevi zadovoljavaju linearnu rekurzivnu relaciju \(B_{n+1} = 6B_{n} - B_{n-1} , n \in \mathbb{N}\). Pokazat ćemo neka svojstva balansirajućih brojeva i ustanoviti povezanost s nekim klasičnim diofanstskim jednadžbama. Također, definirat ćemo kobalansirajuće brojeve malo modificirajući jednadžbu za balansirajuće brojeve i promatrat ćemo njihova svojstva.

Abstract (english)

In this thesis we define balancing numbers\( n \in \mathbb{N}\) as a solution of the Diophantine equation 1 + 2 + ... + (n + 1) = (n + 1) + (n + 2) + ... + (n + r); calling \( r \in \mathbb{N}_0\) the balancer corresponding to \( n \). For example, 1; 6; 35 and 204 are balancing numbers with balancers 0; 2; 14 and 84, respectively. It is interesting that the balancing numbers satisfy linear recurrence relation \(B_{n+1} = 6B_{n}-B_{n-1} , n \in \mathbb{N}\). We show properties of balancing numbers and establish relation to some classic Diophantine equation. Also, we define the cobalancing numbers by slightly modifying equation for balancing numbers and observe their properties.

• Keywords: diofantska jednadžba; balansirajući brojevi; kobalansirajući brojevi
• Keywords (english): Diophantine equation; balancing numbers; cobalancing numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:509781
• 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: 2021-03-11 11:58:49