Upcoming Talks

Zahlentheoretisches Kolloquium

Title: The uniqueness of the extension of infinite two-parameter family of Diophantine triples
Speaker: Dr. Alan Filipin (University of Zagreb)
Date: Freitag, 21. 9. 2018, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract: 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, Togb\' e 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
\begin{align*}
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\}
\end{align*}
with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.

Zahlentheoretisches Kolloquium

Title: The uniqueness of the extension of infinite two-parameter family of Diophantine triples
Speaker: Dr. Alan Filipin (University of Zagreb)
Date: Freitag, 21. 9. 2018, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

Abstract: 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, Togb\' e 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
\begin{align*}
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\}
\end{align*}
with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.