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January 20, 2011. Researchers from Emory, the University of Wisconsin at Madison,
Yale, and Germany's Technical University of Darmstadt discovered
that partition numbers behave like fractals, possessing an
infinitely-repeating structure.
In a collaborative effort sponsored by the American Institute of Mathematics
and the National Science Foundation, a team of mathematicians led
by Ken Ono developed new techniques to explore the nature of
the partition numbers.
"We prove that partition numbers are 'fractal' for every prime.
Our 'zooming' procedure resolves several open conjectures,"
says Ono.
Accompanying this result was another achievement
developing an explicit finite formula for the partition function.
Previous expressions involved an infinite sum, where each term could
only be expressed as an infinite non-repeating decimal number.
Counting the number of ways that a number can be 'partitioned'
has captured the imagination of mathematicians for centuries.
Euler, in the 1700s, was the first to make tangible progress in
understanding the partition function by writing down the generating
series for the function.
These new results
involve techniques which could have applications to other
problems in number theory.
Update 1/21/11: Frank Calegari has provided a shorter proof
of the result of Folsom-Kent-Ono.
This topic is also discussed on MathOverflow,
with an answer by Matt Emerton.
For more background, see the
Emory eScienceCommons blog
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