A Generalization of the Collatz Problem. Building Cycles and a Stochastic Approach

The (3x+1)/2 problem is generalized into the n-furcation problem (l_ix+m_i)/n where i[0, 1, ..., n–1]. It is shown that, under some constraints on l_i and m_i, the main bijection property between the k less significant digits of the seed, written in base n, and the sequence of generalized parities of the k first iterates is preserved. This property is used to investigate a stochastic treatment of ensemble of large value seeds. The bijection property predicts stochasticity for a number of iterations equals to the number of significant digits of the seed. In fact, the stochastic approach is valid for much larger numbers, a property which is more easily shown by using increasing sequences than decreasing ones. Finally, we extend the stochastic approach to cases where the bijection theorem does not hold, introducing the matrix giving the probability that a j number (where j is the last significant digit of this number written in base n) gives a i number iterate.