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Let M = Q( √ −D) be an imaginary quadratic field with ring of integers ZM and let ξ be a root of the polynomial f(x) = x − 2cx + 2x + 2cx + 1, where c ∈ ZM \ {0,±2} and c 6= ±1 if D = 1 or 3. We… (More)

We show that the Diophantine pair $\{1, 3\}$ can not be extended to a Diophantine quintuple in $\mathbb{Z}\left[\sqrt{-2}\right]$. This result completes the work of the first author and establishes… (More)

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine… (More)