Fermatova četvorka

master's thesis

• Author: Lovro Čupić
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Maja Planinić (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Tihomir Vukelja (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Fermatova četvorka je skup \(\{1,3,8,120 \}\) sa svojstvom da je umnožak bilo koja dva njegova elementa uvećan za 1 potpuni kvadrat. Općenito, skupovi od \(m\) prirodnih brojeva s danim svojstvom nazivaju se Diofantove \(m\)-torke. U ovom radu pokazujemo da ako je \(\{1,3,8, d \}\) Diofantova četvorka, tada je \(d = 120\). To su izvorno dokazali Baker i Davenport u [1], gdje su uveli metodu osnovanu na teoriji linearnih formi u logaritmima algebarskih brojeva i metodu redukcije temeljenu na verižnim razlomcima.

Abstract (english)

Fermat’s quadruple is a set \(\{1,3,8,120 \}\) with a property that the product of any two of its elements increased by 1 is a perfect square. Generally, sets of \(m\) positive integers with the given property are called Diophantine \(m\)-tuples. In this thesis, we show that if \(\{1,3,8, d \}\) is a Diophantine quadruple, then \(d = 120\). This was originally proved by Baker and Davenport in [1], where they introduced a method based on the theory of linear forms in the logarithms of algebraic numbers and a reduction method based on continued fractions.

• Keywords: Fermatova četvorka; Baker; Davenport; verižni razlomci
• Keywords (english): Fermat’s quadruple; Baker; Davenport; continued fractions
• Language: croatian
• URN:NBN: urn:nbn:hr:217:359463
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-04 13:04:50