Mathematicians solve the mystery of the ‘impossible’ number – 07/13/2023 – Science

With the help of a supercomputer, researchers manage to calculate the ninth Dedekind number, which for 32 years was considered an unsolvable problem. Numerical sequence was discovered in the 19th century.

After three decades of trying, mathematicians managed to identify the value of a complex number previously considered impossible to calculate. With supercomputers, two groups of researchers unveiled the ninth Dedekind number or D(9) – a sequence of integers along the lines of the better-known prime numbers or the Fibonacci sequence.

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

Until recently, only the eighth Dedekind number was known, unveiled only 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 mathematical problem.

Both studies were submitted to the arXiv preprint server: the first on April 5 and the second on April 6. Despite not yet being peer-reviewed, the two research groups reached the same conclusion – indicating that Dedekind’s ninth number has finally been deciphered.

The ninth Dedekind number or D(9)

The value for the ninth Dedekind number was 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)’s 23 digits.

Each Dedekind number represents the number of possible configurations of a certain type of true-false logical operation in different spatial dimensions. The first number in the sequence, the D(0), represents the zero dimension. Thus, D(9), which represents nine dimensions, is the tenth number in the sequence.

The concept of Dedekind numbers is difficult to understand for those who don’t like mathematics. His calculations are extremely complex, as the numbers in this sequence increase exponentially with each new dimension. This means that they get harder and harder to define, in addition to getting bigger and bigger – which is why the value of D(9) was long considered impossible to calculate.

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

Dedekind numbers

Dedekind numbers are a rapidly growing series of integers. Its logic is based on “monotone boolean functions” (MBFs), which select an output based on inputs consisting of only two possible (binary) states, such as true and false, or 0 and 1.

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

“Basically, you can think of a monotonous Boolean function in two, three and infinite dimensions, as a game with an n-dimensional cube. You balance the cube on a cable and then paint each of the remaining corners white and red”, explains Van Hirtum.

“There is only one rule: you should never put a white corner on top of a red one. This creates a kind of vertical red-white intersection. The object of the game is to find out how many sections there are.”

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

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

The calculation of D(9)

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

The D(9) had almost twice as many digits and was calculated with the Noctua 2 supercomputer at the University of Paderborn. This supercomputer is capable of performing multiple calculations in parallel.

Given the computational complexity of calculating D(9), the team used a P-coefficient formula developed by Van Hirtum’s dissertation advisor, Patrick de Causmaecker. The P-coefficient procedure allowed the calculation of D(9) using a large sum instead of counting 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 just 5.5*10^18, a huge amount. In comparison, the number of grains of sand on Earth equals 7.5*10^ 18, which is not to be sniffed at, but for a supercomputer this operation is quite manageable,” says Van Hirtum.

The researcher, however, believes that the calculation of the tenth Dedekind will require an even more modern computer than the ones currently in existence. “If we calculated it now, processing power equal to the full power of the Sun was required,” Van Hirtum told Live Science. That makes the calculation “virtually impossible”, he added.