%0 Journal Article
%T Collatz Sequences and Characteristic Zero-One Strings: Progress on the 3<i>x</i> + 1 Problem
%A David C. Kay
%J American Journal of Computational Mathematics
%P 226-239
%@ 2161-1211
%D 2021
%I Scientific Research Publishing
%R 10.4236/ajcm.2021.113015
%X <span style=\"font-family:Verdana;\">The unsolved number theory problem known as the 3</span><i><i><span style=\"font-family:Verdana;\">x</span></i><span style=\"font-family:Verdana;\"></span></i><span style=\"font-family:Verdana;\"> + 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 (</span><i><i><span style=\"font-family:Verdana;\">n</span></i><span style=\"font-family:Verdana;\"></span></i><span style=\"font-family:Verdana;\">) and the last (</span><i><i><span style=\"font-family:Verdana;\">m</span></i><span style=\"font-family:Verdana;\"></span></i><span style=\"font-family:Verdana;\">) of a 3</span><i><i><span style=\"font-family:Verdana;\">x</span></i><span style=\"font-family:Verdana;\"></span></i><span style=\"font-family:Verdana;\"> + 1 sequence is analyzed by
means of characteristic zero-one strings. This method is used to achieve some
progress on the 3</span><i><i><span style=\"font-family:Verdana;\">x</span></i><span style=\"font-family:Verdana;\"></span></i><span style=\"font-family:Verdana;\"> + 1 problem. In particular, the</span> long<span style=\"font-family:Verdana;\"><span style=\"font-family:Verdana;\">-</span></span><span style=\"font-family:Verdana;\"><span style=\"font-family:Verdana;\">standing conjecture that nontrivial cycles do not exist is virtually
proved using probability theory.</span></span>
%K Generator
%K Resultant
%K 3&lt
%K i&gt
%K x&lt
%K /i&gt
%K + 1 Cycle
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=112334