Speaker
Description
The set $\{a_1, a_2, \ldots , a_m\}$ in a commutative ring $R$ such that $a_i\ne 0$, $i=1,\ldots,m$ and $a_ia_j+n$ is a square in $R$ for all $1\le i< j\le m$ is called a Diophantine $D(n)$-$m$-tuple in the ring $R$.
Let $N$ be a positive integer such that $4N^2+1=q^j$, $q$ is a prime and $j$ is a positive integer. In this talk, we will discuss the extendibility of the Diophantine $D(-1)$-triple of the form $S_N=\{1,4N^2+1,1-N\}$. More precisely, we will show that the set $S_N$ cannot be extended to a $D(-1)$-quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, with a non-square integer $N$. If $N>1$ is a square, then the set $\{1,4N^2+1,1-N,1+N\}$ is a $D(-1)$-quadruple in the ring $\mathbb{Z}[\sqrt{-N}]$, so in the ring of the Gaussian integers as well.
