Formula for puzzle that captivated Ramanujan

Ramanujan was captivated by the puzzle of how many ways a number can be created by adding together other numbers —a partition of a number is any combination of integers that adds up to that number.

For instance, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1, so the partition number of 4 is 5. However, it becomes more complex with larger numbers and many mathematicians have struggled to find a formula for calculating it. Ramanujan developed an approximate formula in 1918, which 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. He offered no proof but said that these numbers had “simple properties” possessed by no others.

Now, Ken Ono at Emory University in Atlanta, Georgia, and his colleagues have developed a formula that spits out the partition number of any integer. They found “fractal” relationships in sequences of partition numbers of integers that were generated using a formula containing a prime number.

For instance, 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.

Ramanujan’s numbers are the only ones with no fractal behaviour at all. That may be what he meant by simple properties, says Ono.

“It is a privilege to explain Ramanujan’s work. It is something you would never expect to be able to do,” New Scientist quoted Ono as saying.