Prime time at U. for number theory

Theories about numbers aren't just abstract mullings over relationships among digits. They can bring surprising results in technology.

But usually that doesn't happen quickly, warns Peter E. Trapa, associate professor of mathematics at the University of Utah. Last month he discussed "Symmetry, Number Theory and High-Tech Applications" during the U.'s Science at Breakfast program.

Number theory, he told the Deseret Morning News, deals with "just basic questions about numbers." For example, one can list all whole numbers, such as 1, 2, 3, 4, etc. "Among these are certain distinguished ones called prime numbers," he said.

A prime number can't be the product of two smaller numbers. "Twelve is not prime, it's six times two," he said. "Thirteen is prime, it's thirteen times one."

Primes include one, two, three, five, seven, 11, 13 ...

An interesting question concerning primes is how they are spaced? "There's a very precise statement for an estimate and how accurate that estimate is," Trapa said.

The conjecture about the answer was posed by mathematician Bernhard Riemann about a century and a half ago. The formula gives precise answers when tested.

According to the Clay Mathematics Institute of Cambridge, Mass., it has been tested to the first 1.5 billion solutions. But, Trapa said, nobody has yet proven the Riemann Hypothesis.

If anyone ever proves it, that math genius will win the "Millennium Prize" posted by the Clay Institute — which carries a $1 million award.

As the institute points out on the Web at www.claymath.org/millennium, "A proof that is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers."

The practical applications of such theories show up in such originally unexpected places as the design of fast, cheap networks, Trapa said. "There's a lot of work going on at the U." on such topics as harmonic analysis, which could help to speed encoding of data.

"In the past year or so there's been a really spectacular breakthrough" by a colleague of his at the university named Chandrashekhar Khare, a professor in the math department.

"He basically established a profound prediction that ties together seemingly different mathematical theories relating number theory and symmetry," he said.

But don't hold your breath waiting for mathematical theory to translate into technology, Trapa said.

Mathematical abstractions can have a handsome payoff, even greater than the Millennium Prize, but it may take a century or two for technology to catch up with theory. If a person's doctor were "using biological methods that are centuries old, you'd be pretty nervous," he said.

Harmonics theories worked out by pure mathematician Joseph Fourier around 1820 proved to be fundamental to all signal processing, according to Trapa. It is being used today by cell phones.

"There is a trickle-down effect from pure math to technology and applications," Trapa added. But often that takes a long time to materialize.

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E-mail: bau@desnews.com