The convergence classes of Collatz function

The Collatz conjecture, also known as the 3x+1 conjecture, can be stated in terms of the reduced Collatz function R(x)=(3x+1)/2h (where 2h is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we eventually get to 1. Gk, the k-th convergence class, is the set of odd positive integers x such that Rk(x)=1.In this paper an infinite sequence of binary strings sh of length 2⋅3h−1 (the seeds) are defined and it is shown that the binary representation of all x∈Gk is the concatenation of k periodic strings whose periods are sk,…,s1. More precisely where is the substring of length ni that starts in position dk,i in a sufficiently long repetition of the seed si.Finally, starting positions dk,i and lengths ni for which are defined, thus giving a complete characterization of classes Gk.