Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.8

Powers of Two as Sums of Three Lucas Numbers


Bilizimbéyé Edjeou
Université Gaston Berger de Saint-Louis
BP: 234, Saint-Louis
Senegal

Amadou Tall
Université Cheikh Anta Diop de Dakar
BP 5005, Dakar Fann
Senegal

Mohamed Ben Fraj Ben Maaouia
Université Gaston Berger de Saint-Louis
BP 234, Saint-Louis
Senegal

Abstract:

In this paper, we find all positive integer solutions of the Diophantine equation Lk + Ll + Lt = 2d in non-negative integers k, l, t, and d, where (Ln)n≥0 is the Lucas sequence. The tools used to solve our main theorem are linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

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(Concerned with sequence A000032.)

Received August 21 2019; revised versions received August 9 2020; August 26 2020. Published in Journal of Integer Sequences, October 13 2020.

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