### 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.