A generalization of the Collatz problem. Building cycles and a stochastic approach
The well-known and still unsolved \((3x+1)/2\)- or Collatz-problem is generalized into the \(n\)-furcation problem \((l_ix+m_i)/n\)-Problem. It is shown that, under some constraints on \(l_i\) and \(m_i\), the bijection property between the last \(k\) digits written in base \(n\) and the first \(k\) iterates is preserved. This is used to construct arbitrary cycles. Another goal of this paper is to complete the stochastic approach of the Collatz problem. It is conjectured, and tested on a large scale, that the behavior of the iterates started in a large interval around a large value (about \(10^9\)) is equivalent to the behavior obtained by selecting them randomly. In some sense the agreement is excellent.
scientific article; zbMATH DE number 1781680