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Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3 x + 1 Problem)
Author(s) -
Venturini G.
Publication year - 1992
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1992863185
Subject(s) - iterated function , mathematics , generalization , combinatorics , fixed point , periodic point , set (abstract data type) , discrete mathematics , mathematical analysis , computer science , programming language
Given p ∈ N = {1, 2, 3 ...}, as a generalization of the 3 x + 1 problem [7], we study the behavior of the sequences s ( m ) = { m n } n ≥ 0 , m ∈ Z (the set of the integers), defined by the iterative formula and a r = t r /p ( t r ∈ N), and b r are chosen in such a way that m n ∈ 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.