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

2020/06/30 by Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel
Physics and Astronomy · Mathematics · #math.NT

paper · pdf · doi:10.1007/s10474-020-01061-2

Abstract

A set of m distinct positive integers \a₁,… a_m\ is called a D(q)-m-tuple for nonzero integer q if the product of any two increased by q, a_ia_j+q, i≠ 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.

Citations

• There is no Diophantine quintuple

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