Ramanujan’s statement concerned the deceptively simple concept of partitions—the different ways in which a whole number can be subdivided into smaller numbers. […]
[…]One way to think of partitions is to consider how a set of any (indistinguishable) objects can be divided into subsets. For example, if you need to store five boxes in your basement, you can pile them all into a single stack; lay them individually on the floor as five subsets containing one box apiece; put them in one pile, or subset, of three plus one pile of two; and so on—you have a total of 7 options:
5, 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 1+2+2 or 2+3.
Mathematicians express this by saying p(5) = 7, where p is short for partition. For the number 6 there are 11 options: p(6) = 11. As the number n increases, p(n) soon starts to grow very fast, so that for example p(100) = 190,569,292 and p(1,000) is a 32-figure number. […]