Srinivasa Ramanujan

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

I would like (in terms a 12-yr-old can understand) an explanation 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.  

Thanks,
Paul Gee

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 here's something that is fairly easy to
understand:

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(5m+4) is always divisible by 5
   p(7m+5) is always divisible by 7
   p(11m+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/