Simple Example of Ramanujan's WorkDate: 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/ |
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