Certain Diophantine Tuples in Imaginary Quadratic Fields

Abstract

Let $K$ be an imaginary quadratic field and ${\mathcal{O}}_K$ be its ring of integers. A set $\{a_1,a_2,\cdots ,a_m\}\subset {\mathcal{O}}_K\setminus \{0\}$ is called a Diophantine $m$-tuple in ${\mathcal{O}}_K$ with $D(-1)$ if $a_i{a}_j-1=x_{ij}^2$, where $x_{ij}\in {\mathcal{O}}_K$ for all $i,j$ such that $1\le i<j\le m$. Here we prove the non-existence of Diophantine $m$-tuples in ${\mathcal{O}}_K$ with $D(-1)$ for $m>36$.

14 pages,comments are welcome