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Concatenations with Binary Recurrent Sequences
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William D. Banks

Department of Mathematics

University of Missouri

Columbia, MO 65211

USA

Florian Luca

Instituto de Matemáticas

Universidad Nacional Autónoma de México

C.P. 58180

Morelia, Michoacán

México

**Abstract:**
Given positive integers
and ,
we write
for the integer
whose base- representation is the concatenation
of the base- representations of
.
In this paper, we prove that if
is
a binary recurrent sequence of integers satisfying some mild
hypotheses, then for every fixed integer ,
there are at most finitely many nonnegative integers
such that
is
a member of the sequence
. In particular,
we compute all such instances in the special case that , ,
and is the sequence of Fibonacci numbers.

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Received August 30 2004;
revised version received January 13 2005.
Published in *Journal of Integer Sequences* January 14 2005.
Small revisions, January 17 2005.

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