Preface
Motivation
Collatz conjecture
Collatz mapping
The development of this package has been motivated by the famous 3n+1 - Conjecture, which states that iterated application of the Collatz mapping

T: Z -> Z, n |-> n/2 if n even, and n |-> (3n+1)/2 if n odd

T: Z -> Z, n |-> n/2 if n even, and n |-> (3n+1)/2 if n odd
T: \ \mathbb{Z} \longrightarrow \mathbb{Z}, \ \ \ \ n \ \longmapsto \ \begin{cases} \frac{n}{2} & \text{if} \ n \ \text{even}, \\ \frac{3n+1}{2} & \text{if} \ n \ \text{odd} \end{cases}
to any given positive integer eventually yields 1.

This has been conjectured by Lothar Collatz in the 1930s, and is still an unsolved problem today. Replacing the Collatz mapping T by its conjugate under some permutation \sigma of \mathbb{Z} fixing 1 and setwisely fixing the positive integers turns the 3n+1 - Conjecture into the following equivalent conjecture: \forall n \in \mathbb{N} \ \exists k \in \mathbb{N}_0: \ n^{(T^\sigma)^k} = 1. The 3n+1 - Conjecture is true if and only if there is such a permutation \sigma such that T^\sigma maps all integers n > 1 to smaller ones. Hence the problem is to find a suitable normal form for the Collatz mapping T.

Investigating arbitrary permutations of finite sets is both theoretically and computationally a hard problem. It is straightforward to restrict the considerations to permutations which are similar to the Collatz mapping itself. Probably the most natural class of such mappings is the class of residue class-wise affine permutations.

Jeffrey C. Lagarias has written and maintains a commented bibliography , which currently lists 193 references to publications related to the 3n+1 - Conjecture. None of the articles which are referenced there tries to attack the problem by means of group theory, or investigates the structure of groups generated by bijective residue class-wise affine mappings. In fact, the subject residue class-wise affine groups apparently has not been treated anywhere in the literature before.

The author feels that this is a gap which is worth to be filled, and in particular that the group of all residue class-wise affine bijections of the integers is an object of natural interest in its own.

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Purpose of this package

The purpose of this package is to serve as a tool for computational investigations of residue class-wise affine mappings and -groups mainly over the ring of the integers.

Perhaps the only sensible reason for using this package is being fascinated by the mathematical beauty of the objects it helps to investigate.

This manual is pure software documentation, and as such it does not contain any theorems or proofs. In a few places, where this is absolutely necessary for understanding what some function is good for, corresponding mathematical assertions are made. Proofs for all of them as well as a detailed development of the theoretical background for the subject can be found in the author's PhD thesis .

Scope of this package

There are relatively elaborate methods for dealing with residue class-wise affine groups whose elements have a bounded number of different affine partial mappings. These groups have a relatively easy structure. Hence they are called tame. Understanding tame residue class-wise affine groups is a necessary prerequisite for being able to investigate those which are not tame.

The groups which are not tame are much more difficult to examine. Hence these groups are called wild. Nevertheless, some computations can be done also in them -- there is e.g. an algorithm for factoring elements into generators, which usually works reasonably well if the resulting word is not too long. Often it is also possible to get useful information about a wild group by considering its action on finite orbits -- of course provided that such orbits exist.

It cannot be said in a few sentences what can be found out with this package about which mappings or groups. The best way to get an idea about this is to experiment yourself with the examples discussed in this manual and included in the file pkg/rcwa/examples/examples.g. Another useful source of examples is the - function. Often the package does not provide an out-of-the-box solution. At the beginning you will perhaps be bored by extremely long runtimes for seemingly trivial things. But with some experience you will learn to estimate in advance how long something will take and to see why raising some harmlessly-looking mapping to the 20th power would take terabytes of memory, while one can easily find out non-trivial things about some group which looks much more complicate. Provided that you have &LaTeX; and `xdvi` installed, you can nicely display examples of rcwa mappings by repeatedly issueing Display(Random(RCWA(Integers)):xdvi);. Quite often it is possible to find an answer for a given question by using an interactive trial-and-error approach.

The author for example has found with substancial help of this package a non-trivial normal subgroup of the group of all residue class-wise affine bijections of the integers.

Acknowledgements

I would like to thank Bettina Eick for her kind help in trying to make this package and in particular its documentation more useful and more interesting for a larger number of people. Furthermore I would like to thank the two anonymous referees for their constructive criticism and helpful suggestions.

If you use &RCWA; in some work leading to a publication, I ask you to cite it just as you would cite a journal article. I would be grateful for any bug reports, comments or suggestions and of course for reports of results found with the help of this package.

<P align = right> Stefan Kohl </P>
\vfill \noindent Stuttgart, \today \hfill Stefan Kohl