engleski

On extensibility of some parametric families of D(−1)-pairs to quadruples in rings of integers of the imaginary quadratic fields

Let $R$ be a commutative ring. A set of $m$ distinct elements in $R$ such that the product of any two distinct elements increased by $z\in R$ is a perfect square is called a $D(z)$-$m$- tuple in $R$. Let $z=-1$, $R=\mathbb{; ; ; ; Z}; ; ; ; [\sqrt{; ; ; ; - t}; ; ; ; ], t > 0$ and $p$ be an odd prime number. We study the extendibility of $D(-1)$- pairs $\{; ; ; ; 1, p\}; ; ; ; $ and $\ {; ; ; ; 1, 2p^i\}; ; ; ; , i \in \mathbb{; ; ; ; N}; ; ; ; $ to quadruples in $R$. To do it, we study the equation $x^2-(p^{; ; ; ; 2k+2}; ; ; ; +1)y^2=- p^{; ; ; ; 2l+1}; ; ; ; $, $l \in\ {; ; ; ; 0, 1, \dots, k\}; ; ; ; , k \geq 0$ and prove that it is not solvable in positive integers $x$ and $y$.

Diophantine triple ; quadratic field ; Diophantine equation ; Diophantine quadruple

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