Collatz Sequences and Characteristic Zero-One Strings: Progress on the 3x + 1 Problem

Abstract

The unsolved number theory problem known as the 3x + 1 problem involves sequences of positive integers generated more or less at random that seem to always converge to 1. Here the connection between the first integer (n) and the last (m) of a 3x + 1 sequence is analyzed by means of characteristic zero-one strings. This method is used to achieve some progress on the 3x + 1 problem. In particular, the long-standing conjecture that nontrivial cycles do not exist is virtually proved using probability theory.