Simple Example of Ramanujan's Work

Date: 03/28/99 at 18:46:23
From: Paul Gee
Subject: Srinivasa Ramanujan

I would like an explanation, in a manner that a 12 year-old can 
understand, of the mathematical contributions of Srinivasa Ramanujan, 
and why they are significant. If possible, I would also like a simple 
sample problem. I have read many articles about Ramanujan that refer 
to 'elliptical functions' or 'elliptic integrals', 'continued 
fractions', 'infinite series' or 'hypergeometric series', and 
'functional equations of the zeta function'. I do not understand these 
terms, and neither does my older sister, who takes Math Analysis.

Date: 03/29/99 at 12:33:12
From: Doctor Wilkinson
Subject: Re: Srinivasa Ramanujan

This is kind of a hard question to answer, since most of Ramanujan's 
work was pretty advanced. But let's look at this:

Every whole number can be written as a sum of whole numbers in various 
ways.

For example:

  2 = 2 + 0 
    = 1 + 1

  3 = 3 + 0 
    = 2 + 1 
    = 1 + 1 + 1

  4 = 4 + 0 
    = 3 + 1 
    = 2 + 2 
    = 2 + 1 + 1 
    = 1 + 1 + 1 + 1

The number of ways that a whole number n can be written as a sum of 
whole numbers is called the number of partitions of n, and is denoted 
p(n). The first few values are

   p(1) = 1
   p(2) = 2
   p(3) = 3
   p(4) = 5
   P(5) = 7

Ramanujan discovered some amazing divisibility properties of p(n), 
namely:

   p(5n+4)  is always divisible by 5
   p(7n+5)  is always divisible by 7
   p(11n+6) is always divisible by 11

Ramanujan and his friend G. H. Hardy also succeeded in finding an exact
formula for the function p(n).

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/