AI 中文总结
研究关于$D(9)$和$D(64)$丢番图三元组，推广刻画特定$D(l^2)$-丢番图三元组对应佩尔方程解的结果，利用该结果及代数数对数线性形式的界，对特定形式的$D(9)$和$D(64)$-丢番图三元组进行分类。
AI 中文摘要
对于非零整数$q$，一组$m$个不同的正整数$\{a_{1},\dots a_{m}\}$若任意两个数的乘积加上$q$（即$a_{i}a_{j}+q$，$i\neq j$）为完全平方数，则称其为$D(q)$-$m$-元组。因序列某些性质，有许多与斐波那契数相关的$D(q)$-丢番图三元组。Baćić和Filipin的结果刻画了特定形式$D(4)$-丢番图三元组对应的佩尔方程的解。本文推广此结果以刻画满足特定整除条件的$D(l^2)$-丢番图三元组对应的佩尔方程的解。随后利用此结果及代数数对数线性形式的界，对$\{F_{2n+8},9F_{2n+4},F_{k}\}$和$\{F_{2n+12},16F_{2n+6},F_{k}\}$形式的所有$D(9)$和$D(64)$-丢番图三元组进行分类，其中$F_{i}$表示第$i$个斐波那契数。
英文摘要
A set of $m$ distinct positive integers $\{a_{1},\dots a_{m}\}$ 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 Baćić 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 $D(l^2)$-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 $i$th Fibonacci number.
Comments28 pages. This is a pre-print of an article published in Acta Mathematica Hungarica. The final published version is available at https://doi.org/10.1007/s10474-020-01061-2
Journal refActa Math. Hungar., 162 (2) (2020), 483-517
DOI：10.1007/s10474-020-01061-2
