Scientists from the Universities of Paderborn and Leuven have made a groundbreaking discovery in the field of mathematics. They have successfully solved a long-known problem by calculating the ninth Dedekind number, a mathematical sequence of enormous complexity. The exact number, previously believed to be uncomputable due to its size, is 286386577668298411128469151667598498812366.
The scientists used the Noctua supercomputer, along with specialized hardware accelerators, to carry out the calculations. This ambitious project was initially started as a master’s thesis by Lennart Van Hirtum, a computer science student at KU Leuven, who is now a research associate at the University of Paderborn.
The significance of this achievement cannot be understated. Experts around the world have been searching for the exact value of the ninth Dedekind number since 1991. The previous number in the sequence, the eighth Dedekind number, was found in 1991 using a Cray 2 supercomputer, which was the most powerful at the time.
The Dedekind numbers are related to monotone Boolean functions, which can be visualized as a game with an n-dimensional cube. The objective of the game is to count how many different cuts can be made without violating a specific rule. These numbers quickly become enormous, with the eighth Dedekind number already having 23 digits.
To calculate the ninth Dedekind number, the scientists utilized a technique developed by Patrick De Causmaecker, known as the P-coefficient formula. This formula provides a way to calculate Dedekind numbers through a large sum, rather than by counting. While the eighth Dedekind number could be decoded in just eight minutes on a normal laptop, it would take hundreds of thousands of years to calculate the ninth number using the same method.
The main challenge was the exponential growth of the number of terms in the formula. However, by exploiting symmetries in the formula and utilizing application-specific hardware called field programmable gate arrays (FPGAs), the scientists were able to reduce the number of terms to a manageable amount. They used the Noctua 2 supercomputer at the University of Paderborn, which has one of the world’s most powerful FPGA systems.
After several years of development and five months of calculations on the supercomputer, the scientists finally found the ninth Dedekind number on March 8. This achievement marks a major milestone in mathematics and has garnered significant attention from the scientific community.
Lennart Van Hirtum, along with Patrick De Causmaecker, will present their extraordinary success at the University of Paderborn on June 27. Van Hirtum, who is now working on his Ph.D., aims to continue developing the next generation of hardware tools at the Paderborn Center for Parallel Computing. This groundbreaking research opens up new possibilities in the field of mathematics and computational science.