Naslov
On the existence of Diophantine quintuples

Autori
Filipin, Alan ; Cipu, Mihai ; Fujita, Yasutsugu

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
14th Meeting of the Canadian Number Theory Association / - , 2016, 24-24

Skup
14th Meeting of the Canadian Number Theory Association

Mjesto i datum
Calgary, Kanada, 20.06.2016. - 24.06.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Diophnatine m-tuples; hypergeometric method; linear forms in logarithms

Sažetak
A set of m positive integers is called a Diophantine m-tuple if the product of any two of its distinct elements increased by 1 is a prefect square. One of the interesting question is how large those sets can be. There is a folklore conjecture that there does not exist a Diophantine quintuple. In 2004 Dujella proved that there does not exist a Diophantine sextuple and that there are only finitely many quintuples. Recently, there is lot of work from various authors who have improved that result, but the conjecture still remains open. In this talk we give a proof that there does not exist a Diophantine quintuple {;a, b, c, d, e}; such that a < b < c < d < e if a and b are relatively near to each other. Moreover, there is a stronger version of that conjecture, that every Diophantine triple can be extended to a quadruple with a larger element in the unique way. Precisely, if {;a, b, c, d}; is Diophantine quadruple such that a < b < c < d, then d=d_{;+};=a+b+c+2(abc+rst), where r, s and t are positive integers satisfying r^2=ab+1, s^2=ac+1 and t^2=bc+1. In this talk we will also give the proof of that for some families of Diophantine triples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA