Naslov
The extension of the infinite two-parameter family of Diophantine triples

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

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, neobjavljeni rad, znanstveni

Skup
Conference on Diophantine m-tuples and Related Problems II

Mjesto i datum
Hammond (IN), Sjedinjene Američke Države; Westville (NJ), Sjedinjene Američke Države, 15.10.2018. - 17.10.2018

Vrsta sudjelovanja
Plenarno

Vrsta recenzije
Podatak o recenziji nije dostupan

Ključne riječi
Diophantine m-tuples

Sažetak
A set of m positive integers is called a Diophantine m-tuple if the product of any two elements in the set increased by 1 is a perfect square. One of the question of interest is how large those sets can be. Very recently He, Togbe and Ziegler proved the folklore conjecture that there does not exist a Diophantine quintuple. There is also stronger version of that conjecture which states that every Diophantine triple can be extended to a quadruple, with a larger element, in a unique way. That conjecture is still open. In this talk we study the two families of Diophantine pairs and consider their extension. More precisely we prove the mentioned conjecture for the triples $\{;a, b, c\};$, where $a$ and $b$ are positive integers defined by $a=KA^2$, $b=4KA^4+4\varepsilon A$ with $K, A$ positive integers and $\varepsilon \in \{;\pm1\};$ and $c$ is given by $c=c_{;\nu};^{;\tau};$, where $c_{;\nu};^{;\tau};=\frac{;1};{;4ab};\left\{;(\sqrt{;b};+\tau \sqrt{;a};)^2(r+\sqrt{;ab};)^{;2\nu};+(\sqrt{;b};-\tau \sqrt{;a};)^2(r-\sqrt{;ab};)^{;2\nu};-2(a+b)\right\};$, with $\nu$ a positive integer and $\tau \in \{;\pm\};$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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