How a Dresden mathematician won the race for a long-sought number

Dresden mathematician Christian Jäkel spent more than six years searching for the 9th Dedekind number. What he did not know: he was not the only one. When he published his impressive result of the 42-digit number on the preprint server ArXiv on 3 April 2023, it was followed only three days later by a publication from the University of Paderborn presenting the same result. It was a race against time, which Christian Jäkel won due to his quick publication. However, the Paderborn team around computer scientist Lennart Van Hirtum was able to show that they already had the result in March 2023, but were not yet sure about its exactness. In the end, both research groups see their results confirmed by their mutual publications. 

"What is provable should not be believed in science without proof" - is one of the most famous quotes by the German mathematician Julius Wilhelm Richard Dedekind (1831-1916). And so it should come as no surprise that scientists all over the world are working to solve the so-called Dedekind problem, probably the oldest problem in the lattice theory as part of algebra. In 1897, the visionary Dedekind presented a sequence of integers with double exponential growth defined by the number of elements of an algebraic structure. Today, this structure is called a free distributive lattice with n generators. An example from everyday life for the determination of Dedekind's sequence is provided by Manon Bischoff in a Spektrum article from 14 July 2023 [German only]. To this day, there is still no simple formula for calculating the so-called Dedekind numbers d(n). The first four numbers were calculated by Dedekind himself, d(5) by Randolph Church (1940), d(6) by Morgan Ward (1946), d(7) by Randolph Church (1965) and d(8) by Doug Wiedemann (1991). 

The sequence reads: 
d(1) = 3 
d(2) = 6 
d(3) = 20 
d(4) = 168 
d(5) = 7581 
d(6) = 7 828 354 
d(7) = 2 414 682 040 998  
d(8) = 56 130 437 228 687 557 907 788.  
 

To determine d(8), the mathematician Wiedemann had to run the Cray2 supercomputer in 1991 at full speed for a good 200 hours. Since then, the determination of d(9) had been an open challenge, and it was not known whether it would ever be possible to find this number. 

Christian Jäkel came across the problem during his doctoral thesis at the Faculty of Mathematics at the Dresden University of Technology and worked on a solution for six years. With success. "Actually, the Dedekind problem was not the content of his dissertation, but when he stumbled upon the problem, he was fascinated by it. He was obsessed with the idea of determining the ninth Dedekind number and worked on it for six years as part of his PhD. Eventually, he got to grips with the problem in theory and developed a comparably streamlined mathematical method for the computation of the number. I think it is important that science offers enough freedom for such visionary ideas and I admire Christian's ambition, which was crowned with great success in the end," reports Prof. Stefan Schmidt, Christian Jäkel's PhD advisor at the Faculty of Mathematics. 

In the six years of research, Christian Jäkel has set up an algorithm to calculate the 9th Dedekind number. "I worked with a technique we call 'shift' or 'jump'. I applied a shift by 4 and because 5+4 =9, I worked in free distributive lattice with d(5) = 7581 elements. The key aspects are the use of matrix multiplication and symmetries in the free distributive lattice which are determined with techniques of formal concept analysis", the mathematician from Dresden presented his groundbreaking milestone at the ICFCA 2023 (International Conference on Formal Concept Analysis) in Kassel in July. The experts are thrilled by the mathematical sophistication and congratulate the 38-year-old. He has discovered formulas that facilitate efficient calculation of the number on graphics processing units (GPU), resulting in significantly faster runtimes compared to conventional CPU calculations. "During the skiing holiday between Christmas and New Year, I had discovered the approach with matrix multiplication. That was the last piece of the puzzle I had been missing," reports the enthusiastic mathematician. After that, he developed his program, formally wrote down his theory and worked out the necessary proofs. 

"When I had completed everything, I rented eight graphics cards in March and started the calculations," Jäkel describes his final spurt. The eight graphics cards ran for 28 days until Jäkel saw the result in front of him on 3 April 2023. 

The ninth Dedekind number had 42 digits: 

286386577668298411128469151667598498812366. 

At that time, however, Christian Jäkel had no idea that this number had already been determined by another team. On 8 March 2023, computer scientist Lennart Van Hirtum at the University of Paderborn had already come across the number - but with a completely different method. With the help of the Noctua 2 supercomputer located there, a program developed over several years and five months of calculation time, the team came up with the same 42-digit number. Neither did they knew about Christian Jäkel's work. Their method involved the technique developed by Prof. Dr. Patrick De Causmaecker (University of Leuven), known as the P-coefficient formula. With this, d(8) can be decoded in just eight minutes on a normal laptop. (cf. with Christian Jäkel's method, the computation of d(8) can be done within 10 seconds). But the Paderborn team was not sure whether their calculations were correct and so they hesitated to publish the result. Only when they came across Christian Jäkel's preprint publication by chance did they know that their result was correct and set all means in motion to also make their work public as soon as possible, which then happened just three days later. 

When Christian Jäkel saw the publication by Van Hirtum and De Causmaecker, he was very surprised: "I was shocked because I thought it would take five to ten years before another approach could verify the number. To have it verified after such a short time was overwhelming." 

With their almost simultaneous computations, both Christian Jäkel and Lennart Van Hirtum and team were added to Wikipedia as discoverers of the 9th Dedekind number.  

And what’s next? Does Christian Jäkel perhaps also want to determine the 10th Dedekind number? "From d(7) to d(8), and from d(8) to d(9), it took about 30 years in each case. I see no reason why it should go faster from d(9) to d(10). The number d(10) is about 10^82. That corresponds to the number of atoms in the visible universe. Getting there will require new mathematical methods and probably new hardware. I will of course keep a close eye on developments around the Dedekind numbers. There are ideas that I would like to try out, but I'm not hoping to be able to calculate d(10)," he says, describing the challenge. 

About Christian Jäkel:
Christian Jäkel (MSc in Mathematics) is a PhD student at Dresden University of Technology- under the supervision of Prof. Stefan E. Schmidt. His research focuses on optimisation problems in algebraic structures and their behaviour under different product operations. Since 2018, he has been working as a software developer, focusing on pattern recognition in 3D point clouds. 

Original publication: 
Christian Jäkel. A computation of the ninth Dedekind Number. https://arxiv.org/pdf/2304.00895.pdf 

Media inquiries to: 
Christian Jäkel 
Faculty of Mathematics 
TUD – Dresden University of Technology 
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