[IP] the best discoveries don't take long to explain

[David Farber <dave@farber.net>]

------ Forwarded Message
From: Esther Dyson <edyson@edventure.com>
Date: Fri, 01 Apr 2005 20:29:39 -0500
To: <dave@farber.net>
Subject: the best discoveries don't take long to explain

note that this nice little groundbreaker comes in just 10 pages. and note
the last paragraph.

Esther Dyson

>Classic maths puzzle cracked at last
>17:53 21 March 2005
>NewScientist.com news service
>Maggie McKee
>
>A number puzzle originating in the work of self-taught maths genius
>Srinivasa Ramanujan nearly a century ago has been solved. The solution
>may one day lead to advances in particle physics and computer security.
>
>Karl Mahlburg, a graduate student at the University of Wisconsin in
>Madison, US, has spent a year putting together the final pieces to the
>puzzle, which involves understanding patterns of numbers.
>
>"I have filled notebook upon notebook with calculations and equations,"
>says Mahlburg, who has submitted a 10-page paper of his results to the
>Proceedings of the National Academy of Sciences.
>
>The patterns were first discovered by Ramanujan, who was born in India
>in 1887 and flunked out of college after just a year because he
>neglected his studies in subjects outside of mathematics.
>
>But he was so passionate about the subject he wrote to mathematicians in
>England outlining his theories, and one realised his innate talent.
>Ramanujan was brought to England in 1914 and worked there until shortly
>before his untimely death in 1920 following a mystery illness.
>
>Curious patterns
>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.
>
>Patterns everywhere
>Then in the late 1990s, Mahlburg's advisor, Ken Ono, stumbled across an
>equation in one of Ramanujan's notebooks that led him to discover that
>any prime number - not just 5, 7, and 11 - had congruences. "He found,
>amazingly, that Ramanujan's congruences were just the tip of the iceberg
>- there were really patterns everywhere," Mahlburg told New Scientist.
>"That was a revolutionary and shocking result."
>
>But again, it was not clear why prime numbers showed these patterns -
>until Mahlburg proved the crank can be generalised to all primes. He
>likens the problem to a gymnasium full of people and a "big, complicated
>theory" saying there is an even number of people in the gym. Rather than
>counting every person, Mahlburg uses a "combinatorial" approach showing
>that the people are dancing in pairs. "Then, it's quite easy to see
>there's an even number," he says.
>
>"This is a major step forward," Andrews told New Scientist. "We would
>not have expected that the crank would have been the right answer to so
>many of these congruence theorems."
>
>Andrews says the methods used to arrive at the result will probably be
>applicable to problems in areas far afield from mathematics. He and
>Mahlburg note partitions have been used previously in understanding the
>various ways particles can arrange themselves, as well as in encrypting
>credit card information sent over the internet.



Esther Dyson              Always make new mistakes!
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