On solutions of the Diophantine equation Ln+Lm=3a

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DOI:

https://doi.org/10.26637/mjm904/007

Abstract

Let (Ln)n0 be the Lucas sequence given by L0=2,L1=1 and Ln+2=Ln+1+Ln for n0.  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 Ln+Lm=3a 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
  • 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.

Pagdame Tiebekabe and Ismaïla Diouf, Powers of Three as Difference of Two Fibonacci Numbers, $J P$ Journal of Algebra, Number Theory and Applications, 49(2)(2021), 185-196.

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Published

01-10-2021

How to Cite

Tiebekabe, P., and I. Diouf. “On Solutions of the Diophantine Equation Ln+Lm=3a”. Malaya Journal of Matematik, vol. 9, no. 04, Oct. 2021, pp. 228-3, doi:10.26637/mjm904/007.