Modeling of Collatz Sequences in Base-6 Using Markov Chains

Abstract

In this work, we represent and analyze the Collatz sequences in base-6 system and deduce a finite state machine that describes them. This representation of the sequences sheds the light on an interesting insight such as the existence of only four basic types of cycles which constitute any Collatz trajectory. In turn, this allows us to further treat the Collatz process as a Markov chain and, based on the dataset of the first 107 sequences, computationally show that it has a steady state. Moreover, the steady state probabilities theirselves can be directly used to show that the multiplicative factor of a trajectory for any given starting number is less than 1, that is, on average the trajectories tend to converge.

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