Mathematics from the USA and Japan have achieved a major breakthrough in combinatorics by finding the solution to a challenging problem. They have successfully calculated the ninth number of Dedecinda, which represents the number of monotonous Boolean functions with nine variables. Utilizing a supercomputer, they discovered that this number is 2863865768298411128469151667598498812366.
Dedecinda numbers are a rapidly growing sequence of integers named after German mathematician Richard Dedekind, who introduced them in 1897. The number of Dedecinda M (N) is equal to the count of monotonous Boolean functions with n variables. A Boolean function is considered monotonous if changing any input from false to true can only result in a change from false to true and not vice versa. These functions take Boolean variables (1s and 0s) as input and produce a Boolean output.
Calculating Dedecinda numbers poses a significant challenge because there is no known closed formula for their determination. Accurate values are currently only known for N ≤ 9. The first six numbers were obtained in 1940, followed by the seventh and eighth in 1988 and 1990, respectively. The ninth number has remained unknown until now.
To find the elusive ninth number, scientists employed the method of exhaustively generating all possible monotonous Boolean functions with nine variables using the Ku supercomputer. This process required approximately 10 billion hours of processor time and consumed about 500 terabytes of memory. Ultimately, the researchers determined the ninth Dedecinda number to be 28638657682984111128469151667598498812366, which corresponds to the sequence a000372 in the OEIS database.
This discovery holds great theoretical importance in mathematics as Dedecinda numbers are connected to concepts such as antichains in set theory, free distributive lattices, and abstract symmetric complexes. Furthermore, it showcases the capabilities of modern computing technologies in solving complex combinatorial problems.
Although there is no official research report available yet, it is expected to be presented in September at the conference, which will take place in Norway.