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)/2^h (where 2^h 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. G_k, 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 s_h of length 2 · 3^h−1 (the seeds) are defined and it is shown that the binary representation of all x ∈ G_k is the concatenation of k periodic strings whose periods are s_k, . . . , s_1.