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Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

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Given p ∈ N = {1, 2, 3 ...}, as a generalization of the 3x + 1 problem [7], we study the behavior of the sequences s(m) = {mn } n ≥ 0 , m ∈ Z (the set of the integers), defined by the iterative formula and ar = tr/p (tr ∈ N), and br are chosen in such a way that mn ∈ Z for every n. Our aim is to establish when these sequences are divergent, or convergent into a cycle, considering also a fixed point as a cycle. Moreover, the structure of possible cycles is investigated.
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Document Type: Research Article

Affiliations: Scuola Normale Superiore, Pisa

Publication date: April 1, 1992

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