The new finite formula for partition numbers is revealed beginning at 0:25:00 in the video. The explanation of how partition numbers are fractal begins at about 0:47:00.
Mathematician Ken Ono recently presented new breakthroughs in number theory that were centuries in the making. In the above video, you can watch Ono explain the first finite formula to calculate any partition number, and the discovery that partition numbers behave like fractals.
Ono gave the special hour-long lecture for a general audience at Emory on Jan. 21. Although the talk begins with the Count from Sesame Street, it quickly escalates to complex formulas, demonstrating why the simplest problems to state in number theory can be the hardest to solve.
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How do you count a number’s partitions, or the sequences of positive integers that add up to that number? It sounds easy, until you realize that the number 100 has more than 190,000,000 partitions.
If you were to try to write down all the partitions of the number 200, then add up how many there were, you would never finish. It would take you several lifetimes, and you probably wouldn’t arrive at the correct answer anyway.
The Mandelbrot set, above, is the most famous fractal of them all, and demonstrates the endlessly repeating patterns of forms in nature. Ono and his colleagues discovered a new class of fractals, one that reveals the endlessly repeating superstructure of partition numbers. Credit: Wikipedia Commons/Wolfgang Beyer.
Theories for partition numbers may seem esoteric, but they have many real world applications. “The rules of partition function played a role in the early development of computers,” Ono says. “The security of your emails and of your bank card also depend on partition function.”
Ono led a team of researchers to make the recent breakthroughs, including Jan Bruinier of the Technical University of Darmstadt, Amanda Folsom of Yale and Zach Kent of Emory. Their work was funded by the American Institute of Mathematics and the National Science Foundation.
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