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

Abstract

The unsolved number theory problem known as the 3 x + 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 3 x + 1 sequence is analyzed by means of characteristic zero-one strings. This method is used to achieve some progress on the 3 x + 1 problem. In particular, the long - standing conjecture that nontrivial cycles do not exist is virtually proved using probability theory.