Concatenations with Binary Recurrent Sequences
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Given positive integers A1,∙ ∙ ∙ ,At and b ≥ 2, we write A1 ∙ ∙ ∙ At(b) for the integer whose base-b representation is the concatenation of the base-b representations of A1, ∙ ∙ ∙ ,At. In this paper, we prove that if (un)n≥0 is a binary recurrent sequence of integers satisfying some mild hypotheses, then for every fixed integer t ≥ 1, there are at most finitely many nonnegative integers n1,∙ ∙ ∙ ,nt such that │un1 │∙ ∙ ∙│unt│ (b) is a member of the sequence (│un│)n≥0. In particular, we compute all such instances in the special case that b = 10, t = 2, and un = Fn is the sequence of Fibonacci numbers.
William D. Banks and Florian Luca, "Concatenations with Binary Recurrent Sequences", Journal of Integer Sequences, 8 (2005), no.1, Article 05.1.3, 18pp.