Concatenations with Binary Recurrent Sequences
William D. Banks, Florian Luca · 2005
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. 1