September 30, 1998|By Douglas Birch | Douglas Birch,SUN STAFF

PRINCETON, N.J. -- Asked to describe the puzzle that has stumped him for 42 years, Enrico Bombieri takes a piece of chalk out of his pocket and scrawls a lengthy series of numbers and symbols on an upright slab of stone.

The 57-year-old mathematician, standing in a grassy courtyard outside his office at the Institute for Advanced Study in Princeton, scratches furiously for a minute or so. Soon, he rubs out a few numbers with his finger and writes in new ones.

The formula Bombieri works on so compulsively may be the single most important question in mathematics: the 139-year-old conundrum called the Riemann Hypothesis.

"It's a difficult problem," Bombieri says, in a voice that occasionally accelerates into a mumble. "Either you hit a home run, or you're out. No singles, doubles or triples."

For many mere mortals, figuring a batting average qualifies as a math challenge. And not everyone sees the point. Alfred Nobel considered pure math so irrelevant to people's lives that he snubbed it when he set up his famous prizes. (One persistent, but probably apocryphal story has it that Nobel and a mathematician were rivals for the love of a woman.)

Until the recent movies "Good Will Hunting" and "Pi," the pursuit of mathematical proofs has seldom been depicted as a pulse-quickening quest. But for Bombieri and his colleagues, that's exactly what it is. And to them, no problem is as beckoning or as beautiful as the Riemann Hypothesis.

Now, some say, a solution may be on the horizon.

"A number of suggestions, ideas and hints are coming out," Bombieri says, in a conspiratorial tone. "People don't talk too much about it. But there are all these secret ideas out there."

If so, it would mark a major milestone. For generations, the Riemann Hypothesis has lured some very bright people into squandering years in the futile pursuit of a proof.

The English number theorist G. H. Hardy, dreading a voyage across the North Sea early this century, sent a postcard to a friend claiming to have proved Riemann. God, he knew, would not let him die with such a terrible lie on his conscience.

David Hilbert, a German mathematician who died in 1943, once said that if he could wake from the dead in 1,000 years, his first question would be: Has the Riemann Hypothesis been proved?

John Forbes Nash Jr. won the Nobel Memorial Prize in Economic Sciences in 1994 for his contributions to game theory. Back in the late 1950s, the mathematician tackled Riemann and thought he had solved it. A short time later, he decided he was the emperor of Antarctica. He wound up in a mental hospital.

"It's like going to the moon," says Brian Conrey, an Oklahoma State University professor and executive director of the American Institute of Mathematics. "It's the kind of thing that, if somebody from another civilization visited us and we could communicate with them, they would understand it. Maybe they would have solved it."

Puzzle's creator

The creator of this fiendish puzzle was a quiet, pious scientist: Georg Friedrich Bernhard Riemann, born in Hanover, Germany, in 1826, the son of a Lutheran minister. He set out to study theology. But when Riemann was 14, his teacher lent him an 859-page book on number theory. Riemann returned it six days later.

"That was certainly a wonderful book," he reportedly said. "I have mastered it."

While teaching at the University of Gottingen, Riemann devised a new geometry that would later help Einstein build his theory of relativity. But he is best remembered for dashing off a paper in 1859 that included a conjecture about prime numbers. It would become his most celebrated, and frustrating, legacy: the Riemann Hypothesis.

That work still inspires awe. "Riemann's paper is really a flight of the imagination," Bombieri says.

Mathematicians struggle to explain the hypothesis in layman's terms. But one way of looking at it is as a description of a deep order in the near-perfect randomness of the distribution of prime numbers.

Primes are whole numbers that can be divided only by one and themselves. These include 2, 3, 5, 7, 11, 13, 17 and so on. (The number 1 is a special case and not considered prime.) As numbers soar toward infinity, fewer primes appear. But they never seem to vanish altogether.

Harmonic waveform

For centuries, mathematicians have hunted for ways to predict where these primes will occur, like mapping the hidden oases in the trackless desert of whole numbers. So far, they've failed.

Riemann's hypothesis uncovers a pattern in their distribution. When graphed in a certain way, Bombieri points out, their distribution produces a shape called a harmonic waveform.

"To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful," Bombieri once wrote.