Affiliation:
1. Department of Mathematics King Abdulaziz University Jeddah Saudi Arabia
2. Department of Mathematics Yildiz Technical University Istanbul Esenler Istanbul Turkey
Abstract
Abstract
Let (Fn
)
n≥0 and (Ln
)
n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations
ab
and
ba
together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn
= 10
d
Fm
+ Lk
and Fn
= 10
d
Lm
+ Fk
in nonnegative integers (n, m, k), where d denotes the number of digits of Lk
and Fk
, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
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