Ninth Dedekind Number Found

“Richard Dedekind was a 19th-century mathematical giant, responsible for reshaping the field right down to its foundations. He was the first to give a rigorous definition of infinity; he also came up with the definition of the real numbers that form the basis of much of modern mathematics.”

“In 1897, he published an investigation into a certain numerical pattern. That work led him to define a sequence now called the Dedekind numbers, which count structures in a variety of seemingly unrelated mathematical fields. He ended his paper with the observation that “the number of elements contained in these groups seems to grow very rapidly.””

“he figured out the first four terms before giving up. He wasn’t even sure if his calculation for the fourth term in the sequence — 166 — was correct. (He had it right, even though the number is now usually given as 168, taking into account a couple of trivial examples that Dedekind didn’t bother with.)”

“The 5th and 6th terms were calculated in the 1940s, and the 7th in 1965. In 1991, Doug Wiedemann, who worked for the Thinking Machines Corporation, one of the leading supercomputer companies of the time, ran a 200-hour computation to figure out that the eighth Dedekind number, d(8), is 56,130,437,228,687,557,907,788.”

“That’s where things stood until April of this year, when two sets of researchers independently posted their calculations of the ninth Dedekind number, d(9), which is 42 digits long. They used different techniques, and each was unaware of the other. The two papers were posted within three days of each other.”

“There are three main ways to define the Dedekind numbers: as the colors of the corners of an n-dimensional cube; in the language of set theory; and using logic.”

“Regardless of which definition you use, the combinations quickly grow unmanageable. If you were to try to figure out d(9) by brute force, you would use more computer memory than exists on Earth, noted Christian Jäkel, a graduate student at the Dresden University of Technology, who wrote the other April paper.”

“Both papers concluded that d(9) = 286,386,577,668,298,411,128,469,151,667,598,498,812,366.”