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

Abstract

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.