Scientists at Paderborn University and KU Leuven have decoded the mystery of the ninth Dedekind number which has been investigated by mathematicians for almost three decades. The research team arrived at the exact sequence of numbers using the Noctua supercomputer.
Mystery Number Finally Revealed
For 32 years, mathematicians worldwide have been looking for the value of the ninth Dedekind number. The earlier numbers in the series were discovered in 1897 by mathematician Richard Dedekind who was the first to define the problem. Computer science experts, such as Randolph Church and Morgan Ward, also contributed to finding the other numbers.
Understanding the concept of a Dedekind number is a difficult task, let alone working it out. The calculations are so complex because it involves large numbers. Since 1991, calculating the ninth Dedekind number or D(9) has become an open challenge, posing questions whether it will ever be possible to compute this number at all.
The eighth Dedekind number was discovered in 1991 using Cray 2, an instrument considered as the most powerful supercomputer at the time. From this success, experts thought that it should be possible to calculate the ninth number on a large supercomputer. This possibility prompted Lennart Van Hirtum to use the problem as the subject of his master's thesis project while he was still a computer science student at KU Leuven.
Patrick De Causmaecker, Van Hirtum's thesis advisor, developed a technique known as the P-coefficient formula. It offers a strategy in calculating the Dedekind number not by counting but by using a very huge sum. It even decodes D(8) in just eight minutes when calculated on a normal laptop. However, eight-minute calculation for D(8) becomes hundreds of thousands of years for D(9) even if large supercomputers are used.
The problem lies on the fact that the number of terms in the formula are growing incredibly fast. Computing the terms on normal processors is slow, and the current hardware accelerator technology is not efficient for this algorithm.
To solve this problem, the experts used highly specialized and parallel arithmetic units called field programmable gate arrays (FPGAs). While looking for a supercomputer with necessary FPGA cards, Van Hirtum found out that the Noctua 2 computer at the "Paderborn Center for Parallel Computing (PC2)" at the University of Paderborn has the most powerful FPGA systems in the world.
The program ran on the supercomputer for five months after several years of development. On March 8, 2023, the scientists finally found the ninth Dedekind number which was revealed as 286386577668298411128469151667598498812366.
What Is a Dedekind Number?
German mathematician Julius Wilhelm Richard Dedekind used arithmetic concepts in redefining irrational numbers. Unfortunately, he was not recognized in his time, but he influenced modern mathematics by his knowledge of the infinite and the composition of a real number.
Determining the number of monotone Boolean functions of n variables is called Dedekind's problem, and the numbers themselves are called Dedekind numbers. A monotone Boolean function is an increasing function from P(S) which is the set of subsets of S, to {0,1}. The Dedekind number counts the number of elements in a free distributive lattice with n generators, the number of antichains of subsets of an n-element set, or the number of abstract simplicial complexes with n elements.
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