On certain $D(9)$ and $D(64)$ Diophantine triples

Abstract

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 $q$, $a_i{a}_j+q$, $i\ne j$ 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.

28 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