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Mathematicians Unearth the Elusive Ninth Complex Number After Three Decades of Research
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Mathematicians Unearth the Elusive Ninth Complex Number After Three Decades of Research

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After three decades of searching, and with some help from a supercomputer, mathematicians have finally discovered a new example of a special integer called the Dedekind number.

It’s only the ninth of its kind, or D(9), and is calculated to be 286,386,577,668,298,411,128,469,151,667,598,498,812,366 if you’re updating your own records. This 42-digit number follows the 23-digit D(8) discovered in 1991.

The concept of the Dedekind number is difficult for non-mathematicians to understand, let alone solve. The calculations involved are in fact so complex and involve such huge numbers that it was not certain that D(9) would ever be discovered.

“For 32 years, calculating D(9) was an open problem, and it was doubtful whether it was even possible to calculate this number,” says computer scientist Lennart van Hoert from the University of Paderborn in Germany.

In the middle of the Dedekind number are Boolean functions, a type of logic that chooses an output from inputs of only two states, such as true and false or 0 and 1.

Boolean monotonic functions are defined as those that restrict logic in such a way that changing 0 to 1 in the input only changes the output from 0 to 1, not 1 to 0.

Researchers describe it using red and white instead of 1 and 0, but the idea is the same.

“Essentially, you can think of a monochromatic Boolean function with two, three, and infinite dimensions as playing with an n-dimensional cube,” says van Hert, “you balance the cube on one corner, and then color each of the remaining corners either white or red. There is one rule. “Only: the white corner must never be on top of the red corner. This creates a kind of vertical intersection of red and white. The object of the game is to count how many different cuts there are.”

And mathematicians calculate D(1) as 2, then 3, 6, 20, 168…

In 1991, it took the Cray-2 supercomputer (one of the most powerful supercomputers of the time) and mathematician Doug Wiedemann 200 hours to recognize D(8).

D(9) was almost twice as long as D(8) and required a special type of supercomputer: one that used specialized blocks called Field Programmable Gate Arrays (FPGAs) that could perform multiple calculations in parallel. And he led the team to the Noctua 2 supercomputer at the University of Paderborn.

“Solving complex combinatorial problems with FPGAs is a promising field of application, and Noctua 2 is one of the few supercomputers in the world that you can experiment with,” says computer scientist Christian Plessel, head of the Paderborn Center for Parallel Computing (PC2), where Noctua is stored 2. Absolutely.”

More improvements were needed to get Noctua 2 to work with anything. Using symmetry in the formula to make the process more efficient, the researchers gave the supercomputer one huge process involving 5.5 * 10^18 terms (for comparison, the number of grains of sand on Earth is 7.5 * 10^18). .

Five months later, Noctua 2 answered and now we have D(9). At the moment, the researchers have not mentioned D(10), but we can imagine that it could take another 32 years to find it.

There are no paper reports of the research yet, but a paper report is planned to be presented in September at the International Workshop on Boolean Functions and Their Applications (BFA) to be held in Norway.

Source: Science Alert

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