A set of m distinct positive integers $$\{a_{1} , \ldots a_{m}\}$$ is called a $$D(q)-m$$ -tuple for nonzero integer q if the product of any two increased by q, $$a_{i}a_{j}+q, i \neq j$$ is a perfect square. Due to certain properties of the sequence, there are many D(q)-Diophantine triples related to the Fibonacci numbers. A result of Bacic and Filipin characterizes the solutions of Pellian equations that correspond to D(4)-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to D(l2)-Diophantine triples satisfying particular divisibility conditions. Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all D(9) and D(64)-Diophantine triples of the form $$\{F_{2n+8},9F_{2n+4},F_{k}\}$$ and $$\{F_{2n+12},16F_{2n+6},F_{k}\}$$ , where $$F_{i}$$ denotes the ith Fibonacci number.

中文翻译：

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在某些 D(9) 和 D(64) 丢番图三元组上

一组 m 个不同的正整数 $$\{a_{1} , \ldots a_{m}\}$$ 被称为非零整数 q 的 $$D(q)-m$$ -元组，如果任何两个加q，$$a_{i}a_{j}+q，i \neq j$$ 是一个完美的正方形。由于序列的某些性质，有许多与斐波那契数列相关的 D(q)-丢番图三元组。Bacic 和Filipin 的结果表征了对应于某种形式的D(4)-丢番图三元组的Pellian 方程的解。我们概括这个结果是为了表征对应于满足特定可分条件的 D(l2)-丢番图三元组的 Pellian 方程的解。随后，我们在代数数的对数中使用这个结果和线性形式的界限，以便对 $$\{F_{2n+8},9F_{2n 形式的所有 D(9) 和 D(64)-丢番图三元组进行分类+4},