Odd Collatz Sequence and Binary Representations

In this work, we investigate the odd Collatz sequences (odd sequences arising from the 3n + 1 problem) and use binary arithmetic to provide proof of some results in the 3n + 1 problem. Aside from the main result, the paper also provides a new perspective on how to approach the 3n + 1 problem by studying the properties of binary representations of terms in an odd Collatz sequence. This approach is of interest in its own rights. The main result is a generalization of the result of Andaloro [2] on residue class sufficiency sets for the Collatz conjecture to be true: Given a fixed natural number n, the Collatz conjecture is true iff the Collatz conjecture holds for numbers congruent to 1 modulo 2 n. Thus we present a sequence of sets whose set-theoretic limit approaches the set containing only 1. These sets can be chosen to have a natural density that is arbitrarily small. This is an intuitive extension of the result of Andaloro, who proved this for n = 2, 3 and 4. In the past years, sufficiency sets were provided that had similar properties but they usually have a more complex structure [5]. The nearest result was shown around 30 years ago by Korec and Znám [7] who reduced the Collatz conjecture to residue class sufficiency sets that were dependent on primitive roots modulo an arbitrary power of an odd prime. A much simpler sequence of sufficiency sets is presented in this paper.