Collatz Conjecture: Fractals, Networks, And Harder Problems
A child can easily understand the Collatz conjecture, but nobody can prove it. Even after 50 years work by some of the best mathematicians, the problem is unsolved. We look at beautiful fractals and networks that come from the problem.
This is a famous unsolved problem in elementary mathematics. Many famous mathematicians have tried to prove or disprove the conjecture and consequently it has acquired many names (the Ulam conjecture, the 3n + 1 conjecture, Kautani's problem, Thwaites conjecture and the Syracuse problem). Even the great Paul Erdos (who published more mathematics papers than anyone) thought that mathematics was not ready for such a problem.
Despite resisting attacks from so many great mathematicians, the Collatz conjecture is extremely easy to understand. Most ten year old children should be able to grasp it.
The Collatz conjecture says that if you start with any positive whole number, and you keep doing the following operation:
"If the number is even, then change it by dividing it by two, otherwise change it my multiplying it by three and then adding one,"
then you will eventually get to number 1.
A proof or disproof of this open conjecture would be well accepted by the mathematicians of the world.
There are many mathematical objects related to the Collatz Conjectue (the hailstone sequence, wondrous numbers, and Hasse's algorithm), but as far as I know the central problem is still open.
In this video I focus on interesting objects related to the generalizing the Collatz conjecture. In particular, I discuss the Collatz fractal, which was first considered by Letherman, Schleicher, and Wood (1999).
The idea here is to extend the Collatz problem to the complex plane and create a fractal (somewhat akin to Julia sets and Mandlebrot sets) which can be colored to show which complex numbers converge or diverge as the extended Collatz operator is iterated.
I also examine various other closely related problems which I found by exploring networks where vertices represent numbers and edges represent simple operations (like multiplying by three and adding one). Using this novel approach we immediately see that the network associated with the Collatz problem is highly complex, however we also see a great many other very similar looking problems which also generate highly complex networks, and therefore probably also correspond to extremely difficult problems of number theory.
This work reveals many interesting new object that can be generated by `double valued mappings'. It also shows that there are many systems which are very closely related to the Collatz conjecture, but which are also extremely complex.
This work on networks derived from `double valued mappings' is my own research, although it was inspired by the `Multiway Systems Based on Numbers' Wolfram Demonstration by John Cicilio.
http://demonstrations.wolfram.com/Mul...
More details about my work on this see
Experimental Complex Networks, Numbers, Collatz Problem
• Experimental Complex Networks, Number...
and my website
https://sites.google.com/site/richard...
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