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New math theories reveal the nature of numbers

January 24, 2011
Emory University
Mathematicians are unveiling new theories that answer famous old questions about partition numbers, the basis for adding and counting. They have devised the first finite formula to calculate the partitions of any number.


On January 21, Emory mathematician Ken Ono will unveil new theories that answer these famous old questions.


"Our work brings completely new ideas to the problems," says Ono, who explained the findings in a public lecture on the Emory campus. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions."



Child's play



"Partition numbers are a crazy sequence of integers which race rapidly off to infinity," Ono says. "This provocative sequence evokes wonder, and has long fascinated mathematicians."

By definition, partition numbers are tantalizingly simple. But until the breakthroughs by Ono's team, no one was able to unlock the secret of the complex pattern underlying this rapid growth.


"In the mathematical universe, that's like not being able to see further than Mars," Ono says.

A mathematical telescope


"This is like Galileo inventing the telescope, allowing you to see beyond what the naked eye can see, even though the view may be dim," Ono says.

Ramanujan also noted some strange patterns in partition numbers. In 1919 he wrote: "There appear to be corresponding properties in which the moduli are powers of 5, 7 or 11 … and no simple properties for any moduli involving primes other than these three."


In 1937, Hans Rademacher found an exact formula for calculating partition values. While the method was a big improvement over Euler's exact formula, it required adding together infinitely many numbers that have infinitely many decimal places. "These numbers are gruesome," Ono says.

In the ensuing decades, mathematicians have kept building on these breakthroughs, adding more pieces to the puzzle. Despite the advances, they were unable to understand Ramanujan's enigmatic words, or find a finite formula for the partition numbers.

Seeing the forest

Ono's "dream team" wrestled with the problems for months. "Everything we tried didn't work," he says.


"We were standing on some huge rocks, where we could see out over this valley and hear the falls, when we realized partition numbers are fractal," Ono says. "We both just started laughing."

The term fractal was invented in 1980 by Benoit Mandelbrot, to describe what seem like irregularities in the geometry of natural forms. The more a viewer zooms into "rough" natural forms, the clearer it becomes that they actually consist of repeating patterns. Not only are fractals beautiful, they have immense practical value in fields as diverse as art to medicine.

Their hike sparked a theory that reveals a new class of fractals, one that dispensed with the problem of infinity. "It's as though we no longer needed to see all the stars in the universe, because the pattern that keeps repeating forever can be seen on a three-mile walk to Tallulah Falls," Ono says.

Ramanujan's congruences are explained by their fractal theory. The team also demonstrated that the divisibility properties of partition numbers are "fractal" for every prime. "The sequences are all eventually periodic, and they repeat themselves over and over at precise intervals," Ono says. "It's like zooming in on the Mandelbrot set," he adds, referring to the most famous fractal of them all.

The oracle

But this extraordinary view into the superstructure of partition numbers was not enough. The team was determined go beyond mere theories and hit upon a formula that could be implemented in the real world.


"We found a function, that we call P, that is like a magical oracle," Ono says. "I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It's the finite, algebraic formula that we have all been looking for."


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Emory University. "New math theories reveal the nature of numbers." ScienceDaily. ScienceDaily, 24 January 2011. <>.
Emory University. (2011, January 24). New math theories reveal the nature of numbers. ScienceDaily. Retrieved March 28, 2017 from
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