Speaker
Ivan Soldo
(School of Applied Mathematics and Informatics, Josip Juraj Strossmayer University of Osijek)
Description
A set of $m$ non-zero elements of a commutative ring $R$ with unity $1$ is called a $D(−1)$-$m$-tuple if the product of any two of its distinct elements decreased by $1$ is a square in $R$.
We investigate $D(−1)$-tuples in rings $\mathbb{Z}[\sqrt{-k}]$, k ≥ 2. We prove that, under certain technical conditions, there does not exist a $D(−1)$-quadruple of the form $\{a, 2^i
p^j,c,d\}$ in $\mathbb{Z}[\sqrt{-k}]$, with an odd prime $p$, positive integers $a,c,d,j$ and $i\in\{0,1\}$. The main tools in the proof are properties of the related (generalized) Pellian equations.
Author
Ivan Soldo
(School of Applied Mathematics and Informatics, Josip Juraj Strossmayer University of Osijek)
Co-author
Yasutsugu Fujita
(Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba,Japan)