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.
Primary 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)
