Mathematicians who refer to the problem as the 3x+1 problem were never brainwashed by FORTRAN (as I was) into the belief that n, not x, stands for an integer.
Now, in July 2009, I've obtained a detailed mapping of the residue sets which constitute the abstract Collatz predecessor tree. Systematic features of this structure permit development of formulas whose infinite summation indicates the presence of all the odd positive integers therein.; The recursive program which develops details of this structure starting from the root node {5[8]} is effectively an inductive/constructive proof of the Collatz conjecture.
A draft paper (in February 2007) giving the trajectory structure with an ultimate proof of the conjecture and a series of supporting pictures and diagrams are ready for public view and comments. If you print the picture gallery, use minimum borders and colors for best effect. A more recent simplified version of the proof is also provided.
Now in May/June, 2006, I'm trying to streamline, condense, improve the organization of the components, etc. I'll try to do so without worsening the already dense and sometimes confused presentation. I've made a new attempt to produce a very concise presentation of the main arguments I believe will produce a proof of the Collatz conjecture.
As of early April 2005, I've again exhausted what I have to say about the Collatz conjecture, including pages about paradoxes encountered in this work, some discussion of Feinstein's proof of the unprovability of the Collatz conjecture, state transition diagrams to clarify the restrictions on the Collatz transitions, a revised slide show, and numerous additions and corrections. I am ready to claim a proof at this time based on a graph theoretic argument: a binary tree, but no other kind of graph, can be constructed from the nodes representing the odd positive integers. The nodes and their edges are based on the state transition diagram which describes the Collatz trajectories. Showing that the predecessor graph is a tree constitutes a proof of the Collatz conjecture.
If you want a quick trip through my work, look at a somewhat outdated 24 slide show, either serially or through an index, which contains a few pointers to illustrative material. Always use your browser's back button to return to the slide show if you look at some auxiliary material.
I hope someone who can formalize mathematical proofs will see the potential here and take the appropriate set of ideas and sketch or complete a formal proof of the conjecture using them.
You may communicate with me by e-mail at kconrow@ksu.edu. Reports of errors and constructive comments will be particularly welcome.
Glossary of Terms Used
3n+1 Problem Statement and References
All Left Descent Subset Formulas
Represent Disjoint Sets
Brief Overview of Logical Connections
in this Web Site and Proof
Ideas Basic to the Structural View of the Collatz Graph
3n+1 Predecessor Tree, Three Views
The notion of a predecessor tree and details of the general,
binary, and abstract versions of the predecessor tree are presented.
Proof Develops from the Abstract Predecessor
Tree Properties A relatively brief overview is given of the proof to
be presented in these pages.
Development of Formulas Characterizing Every
Left Descent Node
A look at the sets developed during
elaboration of the abstract predecessor tree is presented.
Program Treegrow -- Tracing the Left Descent Assemblies in
the Abstract Tree
The described program produces the
formulas for every element of every left descent assembly up to a user
specified limit and the numeric value of the first instance of all of
them. Treegrow contributed essentially to the cardinality
table, below.
Construction of the Collatz Graph Based on
Residue Sets
A program, rsetprog, similar to treegrow, but based on
residue sets rather than
individual integers, offers an approach to a constructive inductive
proof of the Collatz conjecture.
The Program Rsetprog
A detailed description of the program, its operation, its output,
and my failure to anticipate its behavior correctly are discussed.
(But see the previous index entry.)
Infinite Sums over the Abstract Predecessor
Tree and the Table of Cardinalities
Summation of the densities of the sets
of integers identified in all the left descent assemblies provides for
the totality of the integers.
Ancillary Related Topics
It is pointed out that the method of
construction of formulas of left descent assembly set members results
entirely in disjoint subsets.
Locating an Arbitrary Integer in the Abstract
Predecessor Tree
An algorithm is presented for locating
the position of an arbitrary odd integer within the abstract predecessor
tree.
Programmed Location of an Arbitrary Integer
in the Abstract Predecessor Tree: IDtheLDA
A program using the algorithm of the
above page locating the position of an arbitrary odd integer within the
abstract predecessor tree has been created..
Detailed Paths Including Full Left Descent Assemblies from
any Arbitrary Odd Integer to 1: fullmap
A program tracing the detailed path
from an arbitrary odd integer to 1 shows the details of all left descent assemblies
traversed.
Comparison of Binary Tree Features with
the Graph-Forming Capabilities in the Predecessor Graph
for the (3n+1)/2i Iteration
It is argued from a table of
properties that the individual nodes
within the binary predecessor tree are incapable of forming any graph
other than a binary tree.
Argument via Construction of a Binary Tree
The
individual nodes in the binary predecessor tree have properties which
can produce only a binary tree.
Concise Statement of Arguments Leading to a Proof.
This repeats four major components
of the previous 10 pages selected because they seem most cogent to a
proof of the Collatz conjecture. Readers who already know nearly
everything are advised to start here.
Alternative Proof Involving A State Transition Diagram
State Transition Diagram for Steps Within
One Left Descent Assembly
The allowed transitions within left descents are detailed in support of
the elaboration of the abstract predecessor tree and the consequent
left descent assemblies.
Limits to Conclusions Drawn from the State
Transition Diagram
It has been suggested that the state transition diagram is less than
omnipotent in its support of left descent assemblies and the abstract
predecessor tree.
But Can We All Communicate?
In
communication with several interested investigators, situations have
arisen in which we have had difficulty exchanging ideas relevant to the
following four pages because of different basic assumptions made early
in our independent investigations. These differences are discussed and
mutually justified.
Program Treegrow -- Tracing the Left Descent Assemblies in
the Abstract Tree
The described program produces the
formulas for every element of every left descent assembly up to a user
specified limit and the numeric value of the first instance of all of them.
Program Nbigpath -- Generating the General
Predecessor Tree
Except for time and space imitations,
this program would generate
the entire infinite predecessor tree in the form of a general tree.
Producing "Designer" Path Segments
Less ambitious than nbigpath, two
programs work together to accept a specification of an arbitrary left descent assembly
and produce, then plot, the Collatz trajectory embodying the requested
path.
Some of the Plots of "Designer" Paths
are "Works of Art"
Though not useful in pursuit of a proof,
there is clearly an open field here for producing computer art.
Under construction in September 2005, there is a beginning of sort of a proof outline. It turned out too wordy and unfocused.
State Transition Diagram for
Negative n
A bit of the history of the discovery of the identity of the state
diagram for the Collatz trajectories in the positive and negative
domains is given.
It is full of
mistakes and my state transition diagram is poorly executed compared to
Mensanator's.
Extensions Do Pass Some (Limited) Control
A first hint arose here that certain patterns
of development of the predecessor tree are propagated through iteration sequences
which include extensions.
Why an Arithmetic Argument Fails; Why Twins
(etc.) Arise
Subsets of integers arising in the
stepwise development of the abstract tree have overlapping magnitudes.
The proof that these subsets are all disjoint requires a more
sophisticated argument.
I will modify or add to these pages from time to time as I get feedback from readers or make new discoveries myself.
Let me emphasize that I know no proof with adequate mathematical rigor exists here. That is why I've put these pages up. I hope I'm writing this well enough both to attract other amateurs to the problem and to attract one or more professional mathematicians to the problem of developing a formal proof on the basis of some or all of the structure presented here. At least it should encourage some fresh thinking.