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

摘要

Givenp∈ N = {1, 2, 3 ...}, as a generalization of the 3x+ 1 problem 7, we study the behavior of the sequencess(m) = {mn}n ≥ 0,m∈ Z (the set of the integers), defined by the iterative formulaandar=tr/p(tr∈ N), andbrare chosen in such a way thatmn∈ Z for everyn. 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 inv