What is it about?

Take a positive integer. If it is even, divide it by 2; if it is odd, multiply it by 3 and add 1. Repeat this. The Collatz conjecture asserts that regardless of which number you have started with, you will eventually always end up at 1. Whether this is true is a famous open problem. Solving this problem so far seems hopeless, in spite of hundreds of related articles. In the given article, equivalent group theoretic problems are constructed. The aim is on the one hand to put the Collatz conjecture into a different context, in the hope that eventually techniques from a different area of mathematics can be used to get closer to a solution. On the other, ideas and results in group theory itself are obtained which are motivated by the Collatz conjecture.

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Why is it important?

This paper extends the results obtained in the article "A simple group generated by involutions interchanging residue classes of the integers", which aims at laying the foundations of a theory of residue-class-wise affine groups.

Perspectives

I hope this article will contribute to getting more people work on the subject. My feeling is that many more interesting discoveries related to residue-class-wise affine groups await being made. While the Collatz conjecture is the original motivation for the work, it is not necessarily the main target here. Anyway one might hope to make progress towards the solution of that long-standing problem as well.

Stefan Kohl
University of Saint Andrews

Read the Original

This page is a summary of: The Collatz conjecture in a group theoretic context, Journal of Group Theory, January 2017, De Gruyter,
DOI: 10.1515/jgth-2017-0012.
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