In the face of this number series, human computing power may not be enough

Since the German mathematician Dedekind gave the definition of "Dedekind number" in 1897, mathematicians have only found 9 Dedekind numbers in 2023. This number is 42 digits long and has exhausted the current strongest computing power. When will we find the 10th Dedekind number?

Written by Zhang Hechi

Dedekind numbers are a series of numbers that grow very rapidly. They are named after the German mathematician Richard Dedekind (1831-1916) and were first defined in 1897. In 1991, mathematicians found the eighth Dedekind number, which was 23 digits long. More than 30 years later, in 2023, mathematicians finally calculated the ninth in the sequence, which was 42 digits long. The calculation of Dedekind numbers largely reflects the computing power of today's computers, so it is still unknown when the 10th Dedekind number will be available, and it seems to be a long way off.

What is Dedekind number?

Image from Wikipedia

It is closed under two operations and satisfies the distributive law. Such an algebraic structure is called a (distributive) lattice, and Dedekind, who first studied this structure, is also considered a pioneer of lattice theory. Lattice theory plays a very important role in abstract algebra, logic, theoretical computer science and other fields. However, no one paid attention to Dedekind's lattice theory research after his death. It was not until the 1930s that American mathematician George David Birkhoff rediscovered Dedekind's work in the process of studying universal algebra, and lattice theory officially entered the stage of history. The fifth Dedekind number that Dedekind himself had not found was also found by American mathematician Alonzo Church during this period.

From M(5) to M(8)

Long run in search of M(9)

FPGA stands for Field Programmable Gate Arrays, which is translated into Chinese as field programmable logic gate array. Simply put, it is a semi-customized integrated circuit that can be quickly burned with logic circuits using tools such as logic synthesis, so that the parallel algorithms designed by oneself can be implemented. While Van Hertum was carrying out the theoretical design of the paper, he was looking for a supercomputer with FPGA. The one who finally lent a helping hand was the University of Paderborn in Germany.

The supercomputer Noctua 2 is located in the Paderborn Parallel Computing Center under the University of Paderborn. This computer has one of the largest FPGA systems in the world, and the director of the Parallel Computing Center gladly accepted their request for use after learning the intentions of Van Hertum and Cosmark. For them, using a supercomputer to solve such a complex combinatorial problem is also a very challenging task, which puts forward considerable requirements for the stability and reliability of the system, and is also an excellent opportunity to test the equipment. After several years of development, the program finally started running last year. Van Hertum also graduated from Leuven and entered Paderborn University to pursue a doctorate.

The program ran for 5 months, and on March 8, 2023, the final result was finally obtained.

Arrive at the end point at the same time

References

[1] Jäkel, Christian (2023-04-05). "A computation of the ninth Dedekind Number". arXiv:2304.00895 [math.CO]

[2] Van Hirtum, Lennart (2023-04-06). "A computation of D(9) using FPGA Supercomputing". arXiv:2304.03039 [cs.DM].

[3] Patrick De Causmaecker and Stefan De Wannemacker. On the number of antichains of sets in a finite universe, 2014.

[4] Yusun, TJ: Dedekind numbers and related sequences (2008)

[5] Ninth Dedekind number discovered: Scientists solve long-known problem in mathematics, https://phys.org/news/2023-06-ninth-dedekind-scientists-long-known-problem.html

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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