Contemplating partition (Image: Konrad Jacobs/Oberwolfach Photo Collection)

The first simple formula has been found for calculating how many ways a number can be created by adding together other numbers, solving a puzzle that captivated the legendary mathematician Srinivasa Ramanujan.

The feat has also led to a greater understanding of a cryptic phrase Ramanujan used to describe sequences of so-called partition numbers.

A partition of a number is any combination of integers that adds up to that number. For example, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1, so the partition number of 4 is 5. It sounds simple, yet the partition number of 10 is 42, while 100 has more than 190 million partitions. So a formula for calculating partition numbers was needed.


Previous attempts have only provided approximations or relied on “crazy infinite sums”, says Ken Ono at Emory University in Atlanta, Georgia.

Pattern in partition

Ramanujan’s approximate formula, developed in 1918, helped him spot that numbers ending in 4 or 9 have a partition number divisible by 5, and he found similar rules for partition numbers divisible by 7 and 11.

Without offering a proof, he wrote that these numbers had “simple properties” possessed by no others. Later, similar rules were found for the divisibility of other partition numbers so no one knew whether Ramanujan’s words had a deeper significance.

Now Ono and colleagues have developed a formula that spits out the partition number of any integer. They may also have discovered what Ramanujan meant.

They found “fractal” relationships in sequences of partition numbers of integers that were generated using a formula containing a prime number. For example, in a sequence generated from 13, all the partition numbers are divisible by 13, but zoom in and you will find a sub-sequence of numbers that are divisible by 132, a further sequence divisible by 133 and so on.

Mathematical metaphor

Ono’s team were able to measure the extent of this fractal behaviour in any sequence; Ramanujan’s numbers are the only ones with none at all. That may be what he meant by simple properties, says Ono.

“It’s a privilege to explain Ramanujan’s work,” says Ono, whose interest in partition numbers was sparked by a documentary about Ramanujan that he watched as a teenager. “It’s something you’d never expect to be able to do.”

Trevor Wooley, a mathematician at the University of Bristol, UK, cautions that the use of the term “fractal” to describe what Ono’s team found is more mathematical metaphor than precise description. “It’s a word which conveys some of the sense of what’s going on,” he says.

Wooley is more interested in the possibility of applying Ono’s methods to other problems. “There are lots of tools involved in studying the theory of partition functions which have connections in other parts of mathematics.”