Undeterred, after three decades of research and with the help of a supercomputer, mathematicians have finally discovered a new example of a special integer called the Dedekind number. Only ninth of its kind, or D(9), calculates 286,386,577,668,298,411,128,469,151,667,598,498,812,366 if you update your own records. This 42-digit monster follows the 23-digit D(8) discovered in 1991.

The concept of the Dedekind number is difficult for non-mathematicians to understand, let alone develop. In fact, the calculations were so complex and involved such large numbers that there was no certainty as to whether D(9) would be found.

“For 32 years, calculating D(9) has been an open challenge, and it has been questionable whether it would be possible to calculate this number,” said Lennart Van Hirthum, a computer scientist at the University of Paderborn in Germany. The number was announced in June.

At the heart of the Dedekind number are Boolean functions, or a type of logic that selects an output from an input consisting of only two states, such as true and false, or 0 and 1. Monotonic Boolean functions are functions that restrict logic in this way. Replacing 0 with 1 on the input causes the output to change only from 0 to 1, not from 1 to 0. Researchers explain this by using the colors red and white instead of ones and 0s, but the idea is the same.

“In fact, a monotonic Boolean function in two, three and infinite dimensions can be thought of as a game played with an n-dimensional cube,” Van Geertum said. said. “You balance the cube at one corner and then paint each remaining corner white or red.” “There is only one rule: You must never put the white corner above the red. This creates a sort of vertical red and white cross. The object of the game is to count how many different cuts there are.”

The first few are pretty simple. Mathematicians think D(1) is just 2, then 3, 6, 20, 168.

In 1991, it took the Cray-2 supercomputer (one of the most powerful supercomputers at the time) and mathematician Doug Wiedemann 200 hours to determine D(8). It turns out that D(9) is almost twice as large as D(8) and requires a special kind of supercomputer: one that uses special units called Field Programmable Gate Arrays (FPGAs) that can perform large numbers of calculations in parallel. computer. This led the team to the Noctua 2 supercomputer at the University of Paderborn.

“Solving combinatorial problems with FPGAs is a very promising area of ​​application, and Noctua 2 is one of the few supercomputers in the world that you can try,” says computer scientist Christian Plessl, head of the Paderborn Center for Parallel Computing (PC2). ). ) Where Noctua 2 is stored.

More optimization is needed to make Noctua 2 work. Using symmetry in the formula to make the process more efficient, the researchers gave the supercomputer a very large sum to determine; this contained a total of 5.5*10^18 terms (the number of grains of sand on Earth was estimated to be 7.5*10^). 18, for comparison).

Five months later, Noctua 2 found the answer and now we have D(9). Currently, researchers have not mentioned D(10), but we can estimate that it will take another 32 years to find it. The paper was presented at the International Workshop on Boolean Functions and Applications (BFA) in Norway in September. Source