On solutions of the Diophantine equation Ln+Lm=3aLn+Lm=3a
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https://doi.org/10.26637/mjm904/007Abstract
Let (Ln)n≥0(Ln)n≥0 be the Lucas sequence given by L0=2,L1=1L0=2,L1=1 and Ln+2=Ln+1+LnLn+2=Ln+1+Ln for n≥0n≥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 Ln+Lm=3aLn+Lm=3a in nonnegative integers n,m,n,m, and aa. 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 sequenceMathematics Subject Classification:
11B39 , 11J86- Pages: 228-238
- Date Published: 01-10-2021
- Vol. 9 No. 04 (2021): Malaya Journal of Matematik (MJM)
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