February 24, 2024

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# Supercomputers determine the ‘impossible’ number

Mathematicians equipped with supercomputers have finally determined the value of a complex number that was previously considered “seemingly impossible” to compute.

• The number, known as the “Ninth Dedekind Number” or D(9), is actually the tenth in the sequence.
• Each Dedekind number represents the number of possible configurations of a given type of true-false logical operation in different spatial dimensions.
• The first number in the sequence is D(0), which represents a dimension zero. This is why D(9), which represents nine dimensions, is the tenth number in the sequence.
• Dedekind’s numbers increase exponentially with each new dimension, and it becomes more and more difficult to define.
• Dedekind’s eighth number, which follows the same rules for eight dimensions, was calculated in 1991.
• But because of the leap in computing power needed to calculate the ninth digit, some mathematicians have found it impossible to calculate its exact value.

## The impossible becomes reality

Now, two independent studies from separate research groups — the first uploaded to the arXiv preprint server on April 5th and the second uploaded to the same server on April 6th — have achieved the impossible.

Both studies use a supercomputer but run different software, both studies produced exactly the same number.

Results have not been peer reviewed. But since the studies come to the same conclusion, it is 100 percent certain that the number was deciphered correctly, lead author of the second paper, Lennart van Hertom, a mathematician at the University of Paderborn in Germany, said in an interview with Live Science.

Van Hertom and his colleagues defended their work during a lecture at the University of Paderborn on June 27.

Dedekind numbers were first described by the German mathematician Richard Dedekind in the 19th century. They are related to logical problems known as “monotonic logical functions”.

• Boolean functions are a type of logic that can take only two values ​​as input – 0 (false) and 1 (true) – and output only those two values ​​as output.
• In these functions it is possible to change 0 to 1 on the input, but only if this allows the output to change from 0 to 1, not from 1 to 0. Dedekind numbers are the output of monotonic Boolean functions where the input is a specific spatial dimension.

This concept can be very confusing for non-athletes. Van Hertom showed that it is possible to visualize what is happening by using shapes to represent Dedekind numbers for each dimension.

For example, in the second dimension, the Dedekind number is associated with a square, while the third can be represented by a cube, and the fourth and higher by supercubes.

For each dimension, the vertices or points represent in a certain way the possible configurations of the functions. To find the Dedekind number, it is possible to count the number of times each head of each figure can be colored with one of two colors (in this case, red and white), but provided that one color (in this case, white) cannot be layered on top of the other (in this case, red).

• For zero dimensions, the shape is just a point and D(0) = 2 because the point can be red or white.
• For one dimension, the shape is a line with two points and D(1) = 3 because both points can be the same color or red over white.
• For the two dimensions it is a square and D(2) = 6 because there are now six possible scenarios where there is no white dot over a red dot.
• And for the three dimensions, the shape is a cube, and the number of possible configurations jumps to 20, so D(3) = 20.
• As the number of dimensions increases, the virtual shape turns into an increasingly complex hypercube, with an exponentially greater number of outcomes, van Hertom said.
• The following five Dedekind number values ​​are 68, 7581, 7828354, 2414682040998, and 56130437228687557907788.
• The newly defined value of D(9) is 286386577668298411128469151667598498812366.

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