The mathematical world is buzzing with excitement over the discovery of a new Dedekind number, known as D(9). This notable find was made possible through the power of a supercomputer.
Detailed Discovery
The ninth in its series, D(9), has been computed to be a staggering 286 386 577 668 298 411 128 469 151 667 598 498 812 366. The last discovery of this kind dates back to 1991 with the 23-digit D(8).
About Dedekind Numbers
The world of Dedekind numbers may seem alien to many. These numbers, though deemed difficult to grasp and calculate for non-mathematicians, play a central role in the field. They revolve around Boolean functions, a type of logic which selects an output from a range of two-state inputs.
The concept of monotone Boolean functions further refines this logic. In this system, substituting a 0 for a 1 in an input merely results in the output switching from a 0 to a 1, and not the reverse.
The Calculating Process
The task of calculating Dedekind numbers involves a fascinating process. Corners of an n-dimensional cube are painted with the strict stipulation that no white corner should outrank a red one. The objective then becomes to count the potential different cuts.
Leveraging Supercomputing Power
The process of computation is not for the faint-hearted. The last calculation, D(8), required a Cray-2 supercomputer along with a hefty 200 hours of work. For D(9), with its size being twice that of D(8), a unique supercomputer, capable of executing multiple calculations simultaneously, was demanded.
The Role of Noctua 2
The supercomputer called to duty was the Noctua 2 at the University of Paderborn. With its Field Programmable Gate Arrays (FPGAs), this supercomputer was equipped for the challenge. The researchers harnessed symmetries in the formula to enhance efficiency and tasked the supercomputer with solving a sum consisting of 5.5 * 10^18 terms. A grueling five months later, Noctua 2 successfully computed D(9).
A Major Presentation
The scientific breakthrough was shared with the world at the International Workshop on Boolean Functions and their Applications (BFA) in Norway, leaving the mathematic community in anticipation of what future discoveries might be in store.