Diofantove četvorke sa svojstvom D(n)

master's thesis

• Author: Stela Barović
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Vinko Petričević (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

Neka je \(n\) cijeli broj. Skup cijelih brojeva različitih od nule \(\{a_1, a_2, a_3, a_4\}\) je Diofantova četvorka sa svojstvom \(D(n)\) ako je umnožak bilo koja dva elementa tog skupa uvećan za cijeli broj \(n\) jednak potpunom kvadratu. Dva važna povijesna primjera takvih skupova su \(\{1, 33, 68, 105 \}\) - \(D(256)\)-četvorka (koju je dobio grčki matematičar Diofant Aleksandrijski) te \(\{1, 3, 8, 120\}\) - \(D(1)\)-četvorka (koju je pronašao je francuski matematičar Pierre de Fermat) U ovom radu razmatramo problem postojanja Diofantove četvorke za proizvoljni cijeli broj \(n\). Pokazujemo da Diofantova četvorka sa svojstvom \(D(n)\) postoji ako i samo ako je \(n \not\equiv 2 (mod 4)\), tj. ako i samo ako se \(n\) može prikazati kao razlika kvadrata dva cijela broja, do na konačno mnogo izuzetaka.

Abstract (english)

Let \(n\) be an integer. A set of four nonzero integers \(\{a_1, a_2, a_3, a_4\}\) is a Diophantine quadruple with the property \(D(n)\), or just a \(D(n)\)-quadruple if the product of its any two distinct elements increased by \(n\) is a perfect square. Two significant historical examples of such sets are \(\{1, 33, 68, 105 \}\)- a \(D(256)\)-quadruple (obtained by the Greek mathematician Diophantus of Alexandria) and \(\{1, 3, 8, 120\}\) - a \(D(1)\)-quadruple (found by the French mathematician Pierre de Fermat). In this thesis, we consider the problem of existence of Diophantine quadruples for an arbitrary integer \(n\). We show that there exists a Diophantine quadruple with the property \(D(n)\) if and only if \(n \not\equiv 2 (mod 4)\), i.e. if and only if \(n\) can be represented as a difference of squares of two integers, up to finitely many exceptions.

• Keywords: Diofantova četvorka; Diofant Aleksandrijski; Pierre de Fermat
• Keywords (english): Diophantine quadruple; Diophantus of Alexandria; Pierre de Fermat
• Language: croatian
• URN:NBN: urn:nbn:hr:217:745670
• 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: 2021-03-03 11:55:25