Wednesday, March 23, 2005

Srinivasa Ramanujan Partition Formula Proved For All Prime Numbers



Yup, someone's done it. It's finally been proved. Excerpt:


Ramanujan noticed that whole numbers can be broken into sums of smaller numbers, called partitions. The number 4, for example, contains five partitions: 4, 3+1, 2+2, 1+1+2, and 1+1+1+1.

He further realised that curious patterns - called congruences - occurred for some numbers in that the number of partitions was divisible by 5, 7, and 11. For example, the number of partitions for any number ending in 4 or 9 is divisible by 5.

"But in some sense, no one understood why you could divide the partitions of 4 or 9 into five equal groups," says George Andrews, a mathematician at Pennsylvania State University in University Park, US. That changed in the 1940s, when physicist Freeman Dyson discovered a rule, called a "rank", explaining the congruences for 5 and 7. That set off a concerted search for a rule that covered 11 as well - a solution called the "crank" that Andrews and colleague Frank Garvan of the University of Florida, US, helped deduce in the 1980s.


Quite cool indeed. This is possibly also a major step forward for modern encryption, as most of the algorithms use prime numbers for generating keys. For more information on Srinivasa Ramanujan check out this site.

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