Difficult problem (partitions) studied by Ramanujan solvedClassic maths puzzle cracked at last
17:53 21 March 2005
NewScientist.com news service
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.
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.
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.