On solutions of the Diophantine equation $L_n+L_m=3^a$

Pagdame Tiebekabe; Ismaïla Diouf

doi:10.26637/mjm904/007

Abstract

Let $(L_n)_{n\geq 0}$ be the Lucas sequence given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 0$. In this paper, we are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation $L_n + L_m = 3^{a}$ in nonnegative integers $n, m,$ and $a$. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.

Keywords:

Linear forms in logarithms, Diophantine equations, Perfect powers, Fibonacci sequence, Lucas sequence

Mathematics Subject Classification:

11B39 , 11J86

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Pagdame Tiebekabe Cheikh Anta Diop University, Faculty of Science, Department of Mathematics and Computer science, Laboratory of Algebra, Cryptology, Algebraic Geometry and Applications (LACGAA) Dakar, Senegal.
Ismaïla Diouf University of Kara, Sciences and Tecnologies Faculty (FaST), Department of Mathematics and Computer science, Kara, Togo. PoBOX: 43

Pages: 228-238
Date Published: 01-10-2021
Vol. 9 No. 04 (2021): Malaya Journal of Matematik (MJM)

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Pagdame Tiebekabe and Ismaïla Diouf, On solutions of the Diophantine equations $F_{n_1}+F_{n_2}+F_{n_3}+$ $F_{n_4}=2^a$, Journal of Algebra and Related Topics, Accepted to On-line Publish, 2021.

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