September 28, 2023

# Mathematicians solve the ‘impossible’ number puzzle | world

After three decades of trying, mathematicians have managed to determine the value of a complex number that was previously considered impossible to compute. Using supercomputers, two groups of researchers have revealed the ninth Dedekind number, or D (9) – a series of integers along the lines of known primes or the Fibonacci series.

Among the many mysteries of mathematics, Dedekind’s numbers, discovered in the 19th century by German mathematician Richard Dedekind, have captured the imagination and curiosity of researchers over the years.

Until recently, only Dedekind’s number eight was known, and it was only unveiled in 1991. But now, in a surprising turn of events, two independent research groups from the Catholic University of Leuven in Belgium and the University of Paderborn in Germany have achieved the unthinkable and solved the problem. sports.

Both studies were submitted to the arXiv preprint server: the first on April 5 and the second on April 6. Although not yet peer-reviewed, both research groups have come to the same conclusion – suggesting that Dedekind’s ninth number has finally been decoded.

## Dedekind’s ninth number, or D (9).

The value of the ninth Dedekind number is calculated to be 286,386,577,668,298,411,128,469,151,667,598,498,812,366. D(9) has 42 digits compared to D(8) which has 23 digits.

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, D(0), represents the zero dimension. So D(9), which represents nine dimensions, is the tenth number in the sequence.

The concept of Dedekind numbers is hard to understand for those who don’t like mathematics. His calculations are very complex, as the numbers in this sequence increase exponentially with each new dimension. This means that it gets harder and harder to quantify, as well as it gets bigger and bigger – which is why the value of D(9) has long been seen as impossible to compute.

“For 32 years, calculating D(9) was an open challenge, and it was questionable whether it was ever possible to calculate this number,” says computer scientist Lennart Van Hirtum of the University of Paderborn, author of one of the studies.

Dedekind numbers are an increasing series of integers. Its logic is based on “Montonic Boolean Functions” (MBFs), which select an output based on inputs that consist of only two possible (binary) states, such as true and false, or 0 and 1.

Boolean unary functions constrain logic in such a way that changing the number 0 to 1 on only one input causes the output to change from 0 to 1, not from 1 to 0. To illustrate this concept, researchers use red and white, instead of 1 and 0 , although the basic idea is the same.

“Essentially, you can think of a monotonous logical function in two, three and infinite dimensions, like a game with a cube of n dimensions. You balance the cube on a cable and then paint each of the remaining corners white and red,” van Hertom explains.

“There is only one rule: you should never place a white corner on top of a red corner. This creates a kind of vertical red-and-white cross. The object of the game is to see how many divisions there are.”

Thus, the Dedekind number represents the maximum number of intersections that can occur in a cube of n dimensions that satisfies the rule. In this case, the n dimensions of the cube correspond to the Dedekind number n.

For example, Dedekind’s eighth number has 23 digits, which is the maximum number of different divisions that can be made in an eight-dimensional cube that satisfies the rule.

In 1991, the Cray-2 supercomputer (one of the most powerful computers of the time, but less powerful than a modern smartphone) and mathematician Doug Wiedemann took 200 hours to calculate D(8).

D(9) has almost twice as many digits and was calculated using the Noctua 2 supercomputer at the University of Paderborn. This supercomputer is capable of performing multiple mathematical operations at the same time.

Due to the computational complexity of calculating D(9), the team used the P coefficient formula developed by Van Hirtum’s thesis advisor, Patrick de Causmaecker. Doing the modulus P allows D(9) to be computed using a large sum instead of calculating each term in the series.

In our case, taking advantage of the symmetries of the formula, we were able to reduce the number of terms to only 5.5 * 10^18, which is a huge amount. By comparison, the number of grains of sand on Earth is 7.5 * 10^18, which is not something to sniff out, but to a computer However, this process is completely manageable,” says van Hertom.

However, the researcher believes that Dedekind’s tenth account requires a more modern computer than the ones currently in existence.

“If we calculate it now, then a processing power equal to the full power of the sun will be required,” van Hertom told the portal. Science lives. He added that this makes computation “almost impossible”.