On the Collatz general problem $qn+1$

Abstract

In this work the generalized Collatz problem $qn+1$ ($q$ odd) is studied. As a natural generalization of the original $3n+1$ problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number theory and dynamical systems, general properties are established, such as the existence of finitely many periodic sequences for each $q$. In particular, when $q$ is a Mersenne number, $q=2^p-1$, there only exists one such cycle, known as the trivial one. Further analysis based on a probabilistic model shows that for $q=3$ the asymptotic behavior of all sequences is always convergent, whereas for $q\geq 5$ the asymptotic behavior of the sequences is divergent for almost all numbers (for a set of natural density one). This leads to the conclusion that the so called Collatz Conjecture is true, and that $q=3$ is a very special case among the others (Crandall conjecture). Indeed, it is conjectured that the general problem $qn+1$ is undecidable.