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

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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 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(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.

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The authors would like to thank Dr. Claude Levesque. His advice on this paper was invaluable. The authors thank the referee for a careful reading of the manuscript and for comments which improved this paper.

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Correspondence to S. Earp-Lynch.

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The third author was supported in part by NSERC.

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Earp-Lynch, B., Earp-Lynch, S. & Kihel, O. On certain D(9) and D(64) Diophantine triples. Acta Math. Hungar. 162, 483–517 (2020).

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