On certain D (9) and D (64) Diophantine triples
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ON CERTAIN D(9) AND D(64) DIOPHANTINE TRIPLES B. EARP-LYNCH, S. EARP-LYNCH∗ and O. KIHEL† Department of Mathematics, Brock University, Ontario, Canada L2S 3A1 e-mails: [email protected], [email protected], [email protected] (Received November 30, 2019; revised April 19, 2020; accepted April 26, 2020)
Abstract. A set of m distinct positive integers {a1 , . . . am } is called a D(q)m-tuple for nonzero integer q if the product of any two increased by q, ai aj + 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 Ba´ci´c 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 {F2n+8 , 9F2n+4 , Fk } and {F2n+12 , 16F2n+6 , Fk }, where Fi denotes the ith Fibonacci number.
1. Introduction A set of m (integer) elements {a1 , . . . , am } is called an (integer) Diophantine m-tuple with property D(l), or alternatively a D(l)-m-tuple, provided the product of any two different elements increased by l, ai aj + l where i �= j, is an integer square. Similar sets have been studied for centuries. Fermat showed that the set {1, 3, 8} can be extended from a D(1)-triple to a D(1)quadruple with the additional element 120. Baker and Davenport [2] showed that 120 is in fact the only integer that extends this set from a D(1)-triple to a D(1)-quadruple. This was also the first appearance of the method that came to be known as Baker–Davenport Reduction. He, Togb´e and Ziegler [7] proved that there does not in fact exist a D(1)-quintuple. D(l)-m-tuples where l = −1 and l = 4 have also been studied (see [10], [1], [6] and [8] and ∗ Corresponding
author. third author was supported in part by NSERC. Key words and phrases: Diophantine triple, Pell’s equation, Fibonacci number, linear form in logarithms. Mathematics Subject Classification: 11D09, 11D45, 11B37, 11J86. † The
c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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B. B. EARP-LYNCH, EARP-LYNCH, S. S. EARP-LYNCH EARP-LYNCH and and O. O. KIHEL KIHEL
[16]) as have further cases, though all less extensively than the case when l = 1. The D(1)-triple {1, 3, 8} is an example of a D(1)-triple of Fibonacci numbers. Let Fn denote the nth Fibonacci number, and let F1 = F2 = 1. Using the recurrence relation, Fn+1 = Fn + Fn−1 , the sequence of Fibonacci numbers is seen to be as follows, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . . The Fibonacci numbers obey what is called Binet’s formula, Fn =
αn − αn √ 5
√
√
where here α = 1+2 5 and α is its conjugate, α = 1−2 5 . Note that the D(1)triple {1, 3, 8} consis
