Collatz conjecture

The winter term is almost over and I have seen a lot of new mathematics. I have no more compulsory subjects and so I have been able to choose. I am now studying rings and modules, topology, integration and statistics.

There is a famous open problem in mathematics: take a positive integer n. If n is even then divide it by two, if it is odd then take 3n+1 instead. If we now repeat this process a hypothesis states that the resulting sequence always goes through number one. This was first formulated by Lothar Collatz in 1937, hence it is called the Collatz conjecture.

The formulation of the problem is so very simple almost anyone can understand it. Its seeming harmlessness invites us to generate a few numbers by hand. A short experiment, however, shows us that our sequences behave non-trivially and chaotically. If we, for instance, start from number 27, we will go through 9232 before hitting 1. Suspected patterns usually melt away sooner or later.

This hypothesis has been verified computationally for numbers approximately up to one quintillion, but otherwise, we have only speculations. The proof is non-existent. Much has been said about the difficulty of this problem. In fact, the atmosphere is such that we may not yet be ready for this problem. In other words, it is out of reach for present day mathematics.

If you are asking me, why I have dedicated three paragraphs to this remarkable puzzle, it is because, in only a few months, I will be concerning myself with a topic somewhat related. I have been given the opportunity to stay at the university over the summer to make a research project in the area of computational number theory, analysis and dynamical systems, in which I will be studying fractals constructed in analogous ways.

A significant aspect of the project will be the computational part, in which I will use my experience from last year’s computational projects as well as from my summer internship in Air Bank. I also hope that as a by-product of my work I will create some aesthetically appealing images of fractals and I cannot wait to start working.

Until then I will have preparatory reading to go through. Also, I still need to learn number theory next term.


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