Math Puzzlers Here For Prime Time
Seattle Times Science Reporter
When the British mathematician G.H. Hardy faced a stormy sea passage from Scandinavia to England early this century, he sent a postcard to a colleague with the message, "Have proved the Riemann Hypothesis."
Hardy reasoned that God would not let him drown with undeserved credit for solving what is arguably the world's toughest and most important math problem. Besides, his postcard would torment future generations of scholars unless he lived to retract it.
An estimated 150 of the nation's brightest mathematicians are in Seattle this week trying to do what Hardy claimed: prove that the German mathematician Georg Friedrich Bernhard Riemann was correct in 1859 when he proposed a formula describing the distribution of prime numbers.
While perhaps less famous than Fermat's Last Theorem, which was a 356-year-old mystery when it was solved in 1993, Riemann's is more celebrated among mathematicians.
Its proof, said Princeton mathematician Peter Sarnak - a neighbor to Andrew Wiles, who solved Fermat - could shed light on 300 other math puzzles. It could possibly contribute to secret code-making and breaking, physical theories about the universe and other fields.
There would also be a certain measure of fame for the discoverer. It took Wiles eight years to crack Fermat - a deceptively simple proposition that if n is a whole number larger than 2, there is no solution to the problem x to the nth power plus y to the nth power is equal to z to the nth power.
Afterward, Wiles ended up in People magazine and turned down an offer to model Gap jeans.
What Wiles accomplished with solitary brilliance, a new organization called the American Institute of Mathematics (AIM), is trying to achieve with a group effort.
In calling together a symposium on Riemann at the University of Washington - the meeting follows on the heels of last weekend's Seattle Mathfest, which drew 1,000 people - AIM hopes thinkers from both math and physics will inspire one another.
The symposium, which ends tomorrow, is taking place during the 100th anniversary of the proof of the Prime Number Theorem. It started with a bang Monday when Atle Selberg got an unusual standing ovation (mathematicians are not the most demonstrative bunch) for a brilliant lecture on the history of prime-number theory.
Prime numbers are numbers that can be evenly divided only by themselves or one. Examples are 2, 3, 5, 7, 11 and so on. As numbers grow in size, primes decrease in frequency. At first glance their distribution seems random, but AT&T mathematician Andrew Odlyzko pointed out that one of the purposes of mathematics is "a study of patterns in the complexity we see around us."
Riemann proposed a formula to predict where primes would fall in an infinite progression of numbers, a formula that includes a certain measure of randomness, or error.
So far it works. Odlyzko has used computers to show Riemann's hypothesis works out to the first 1.5 billion or so possibilities.
But while computers have so far failed to show Riemann was wrong, they can never prove him right: Somewhere out in infinity, a prime theoretically could still violate his hypothesis. If mathematicians could come up with a proof showing why Riemann must be correct, it would provide an insight into all kinds of abstract and practical math problems.
One problem is encryption, or encoding information so it can't be read. This is important not only for national security but for such things as electronic banking transactions.
Another is the field of topology, or applying mathematics to problems in three- or four-dimensional space that could explain the fundamental structure of the universe.
Mathematics has been vital in modern technology, from development of computers to long-distance phone calls. The next time you fly, you can thank complex-number theory for the fact the airplane doesn't fall down: The theory was harnessed in the 1920s to explain air flow across airplane wings.
Beyond that, there is a mystical challenge in tackling problems such as Riemann's.
"A lot of religious feeling drives people to mathematics, looking for patterns," Odlyzko said.
Abstract mathematical discoveries have again and again proved to be "amazingly relevant" in describing what goes on in the real world, he explained. Many mathematicians feel they discover in math fundamental truths about the universe.
Solving Riemann's hypothesis is also an irresistible challenge.
"This is important because the giants of our field have worked on it," said Jerry Alexanderson, incoming president of the Mathematical Association of America.
And they have yet to solve it.
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