It was proudly reported earlier today that mathematics' most famous and challenging problem has been solved. A team from the UK's University of Cambridge, using a combination of classical maths and Generative Artificial Intelligence (Gen AI) have demonstrated that the Collatz Conjecture (also known as the 3n+1 problem) has been proved once and for all, using a new dynamic algorithm, similar to the type used by the Chess-playing program, Alpha Zero.
The problem is easy to state, but has confounded mathematicians for almost 100 years: take any number, and if it's odd, then multiply by three and add one. If it's even, divide by two. Take this number, and repeat the operation: if odd, then multiply by three and add one; if even, divide by two. Continue to repeat this operation, and eventually, you reach 1. (1 *3) +1 = 4, 4 /2 = 2, 2/2 = 1.
Mathematically:
The team from Cambridge's Department for Applied Mathematics and Theoretical Physics worked on devising an algorithm that was able to overcome the Collatz Conjecture's key challenges. Instead of trying to unpick the chaotic nature of the Conjecture's sequence, they embraced this using their dynamic Gen AI model. Previously, the challenge of the Collatz Conjecture lay in its number sequence, which can grow to immense sizes swiftly, only to diminish just as quickly. However, when they programmed their AI algorithm to map every integer in a variable 1-4 dimensional space, and plot each term in all sequences in a four-dimensional matrix, they uncovered a spherical symmetry that they had not expected. As all real numbers are contained within this four-dimensional hypersphere, the team were able to prove the Collatz Conjecture for all real positive integers.
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| A 3-D visualization of the 4-D Collatz Conjecture solution University of Cambridge |
As the conjecture’s proof is tied to other mathematical domains, such as number theory and dynamical systems, it is expected that proving it will have far-reaching consequences in these areas, necessitating a profound review of these disciplines.
A second visualisation of the 4-D Collatz Conjecture solution showing a different 'shadow'
University of Cambridge
The team have not yet shared full details of how the proof works, but they explained that they mapped all known sequences into the 1-4 dimensional space, and the AI algorithm then arranged them in a way that would maximise their spatial symmetry. The next step was then to map all the odd, so-called 'April' numbers and connect them to the even numbers. By demonstrating that any odd number would always eventually path to another point on or within the same hypersphere, they were able to prove that all numbers eventually path to 1. The algorithm was able to plot 'shadows' of this in 3D, and the visualizations have been as beautiful as they have symmetrical.
A close up of a 3-D shadow of the 4-D solution, showing the connections between the real integers in Collatz sequences
University of Cambridge
The team plan to publish full details of their findings and proof in an upcoming issue of the Journal of the European Mathematical Society, following a thorough peer review. I will provide more updates as I find them; I have my own series of articles here on this blog on the Collatz Conjecture, and the variations 5n+1, 3n+3 (which is wild) and 3n+5 (which grows very rapidly).
