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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