Researchers from Emory University, the University of Wisconsin Madison, Yale, and the Technical University of Darmstadt in Germany have discovered that partition numbers behave like fractals, possessing an infinitely-repeating structure.
The partition number P(N) of an integer N is the number of distinct ways in which N can be written as a sum of positive integers. For instance, 6 = 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, and 1+1+1+1+1+1, so that P(6) = 11. P(N) grows very rapidly with N. For instance, P(100) = 190,569,292.
Partition numbers have captured the imagination of mathematicians since the time of Euler in the 1700s. In the 20th century, the great Indian mathematician Ramanujan discovered an approximation formula for partition numbers in 1919, but before he could fully develop his theory he died in 1920, at the age of 32.
Inspired by Ramanujan’s work, and also by the work of others working on the problem since 1920, Ken Ono of Emory University, working with Amanda Folsom and Zachary Kent, found an intriguing fractal-like pattern to the set of partition numbers. In a separate paper co-authored by Jan Hendrik Bruiner of Germany, Ono found a new explicit formula for the partition function.
Among other things, the work of these researchers has removed any hope that partition numbers could be used as a scheme to encrypt computer data. As Ono explained, “Nobody’s ever going to do that now, since we now know partition numbers aren’t random.” He added, “They’re completely predictable and we should no longer pretend they’re mysterious.”
For additional details see an interest article by Dave Mosher on the Wired website:
Wired article
and also the following announcement from the American Institute of Mathematics, which includes links to the actual mathematical papers:
AIM site.
The above post was adapted from material in these two sources.