Pitagorine trojke i neke slične diofantske jednadžbe

master's thesis

• Author: Martina Varga
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pitagorine trojke su rješenja diofantske jednadžbe \(x^2+y^2=z^2\). Nakon kratkog povijesnog pregleda, u radu obrađujemo tri dokaza Euklidove formule za generiranje primitivnih Pitagorinih trojki. Osim navedene, proučavamo neke slične diofantske jednadžbe. Pitagorine četvorke su rješenja diofantske jednadžbe \(x^2+y^2+z^2=w^2\). Zatim prikazujemo parametrizirane formule za generiranje primitivnih Pitagorinih četvorki. Blizu Pitagorine trojke su rješenja diofantske jednadžbe \(x^2+y^2=z^2-1\), dok su skoro Pitagorine trojke rješenja diofantske jednadžbe \(x^2+y^2=z^2+1\). Pokazujemo vezu blizu i skoro Pitagorinih trojki s prethodnim jednadžbama te njihov međusobni odnos. Osim toga, u radu navodimo korake za generiranje skoro Pitagorinih trojki.

Abstract (english)

Pythagorean triples are solutions of the Diophantine equation \(x^2+y^2=z^2\). After a short historical review, we elaborate three proofs of Euclid’s formula for generating primitive Pythagorean triples. In addition, we study some similar Diophantine equations. Pythagorean quadruplets are solutions of the Diophantine equation \(x^2+y^2+z^2=w^2\). We present parameterized formulas for generating primitive Pythagorean quadruplets. Nearly Pythagorean triples are solutions of the Diophantine equation \(x^2+y^2=z^2-1\), while almost Pythagorean triples are solutions of the Diophantine equation \(x^2+y^2=z^2+1\). We show a close relationship between nearly and almost Pythagorean triples and their relation with the previous equations. In addition, we outline the steps for generating almost Pythagorean triples.

• Keywords: diofantske jednadžbe; Pitagorine trojke; Pitagorine četvorke
• Keywords (english): Diophantine equation; Pythagorean triples; Pythagorean quadruplets
• Language: croatian
• URN:NBN: urn:nbn:hr:217:787487
• 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; Embargo expiration date: 2019-10-03
• Created on: 2019-07-03 10:01:40