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With the help of a supercomputer, researchers are able to calculate the ninth Dedekind number, which was considered an unsolvable problem for 32 years. The sequence of numbers was discovered in the 19th century.
After three decades of trying, mathematicians managed to find the value of a complex number that was previously considered impossible to calculate. Using supercomputers, two groups of researchers unveiled the ninth Dedekind number, or D(9) a sequence of integers modeled after the more familiar prime numbers or Fibonacci sequence.
Among the many mysteries of mathematics, the Dedekind numbers, discovered by German mathematician Richard Dedekind in the 19th century, have captured the imagination and curiosity of researchers over the years.
Until recently, only the eighth Dedekind number was known, which was only revealed in 1991. But now, in a surprising twist, 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 math problem.
Both studies were submitted to the preprint server arXiv: the first on April 5th and the second on April 6th. Although it hasn’t been peerreviewed yet, the two research groups came to the same conclusion suggesting 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) which has 23 digits.
Each Dedekind number represents the number of possible configurations of a particular type of truefalse logical operation in different spatial dimensions. The first number in the sequence, D(0), represents zero dimension. Thus, D(9), representing nine dimensions, is the tenth number in the sequence.
The concept of Dedekind numbers is difficult to grasp for those who don’t like math. Its calculations are extremely complex, as the numbers in this sequence increase exponentially with each new dimension. This means that they are not only getting larger, but also increasingly difficult to define which is why the value of D(9) was long thought to be unpredictable.
“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 from 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 “Monotonic Boolean Functions” (MBFs) that select an output based on inputs consisting of only two possible (binary) states, such as B. true and false or 0 and 1.
Monotonic Boolean functions constrain the logic so 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 of 1 and 0 instead, although the underlying idea is the same.
“Basically, a monotonic Boolean function in two, three, and infinite dimensions can be thought of as playing a game with an ndimensional 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 redwhite intersection. The aim 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 ndimensional 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. This is the maximum number of distinct sections that can be created in an eightdimensional cube that satisfies the rule.
The computation of D(9)
In 1991, it took a Cray2 supercomputer one of the most powerful of its time, but less powerful than a modern smartphone) and mathematician Doug Wiedemann 200 hours to compute D(8).
The D(9) had almost twice as many digits and was calculated with the supercomputer Noctua 2 from the University of Paderborn. This supercomputer is able to carry out several calculations in parallel.
Given the computational complexity of computing D(9), the team used a Pcoefficient formula developed by Van Hirtum’s PhD advisor Patrick de Causmaecker. The Pcoefficient method allowed D(9) to be computed using a large sum instead of counting each term in the series.
“In our case, by exploiting the symmetries of the formula, we were able to reduce the number of terms to just 5.5*10^18, an enormous amount. In comparison, the number of grains of sand on Earth is 7.5*10 ^ 18, which is not to be scoffed at, but for a supercomputer this operation is quite feasible,” says Van Hirtum.
However, the researcher believes that calculating the tenth Dedekind will require an even more modern computer than those currently in existence. “If we calculated it now, it would require computing power equal to the full power of the sun,” Van Hirtum told Live Science. This makes the calculation “virtually impossible,” he added.