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.

The hypothesis can be used to analyze concert hall acoustics. Some physicists think it can help explain the origins of chaos, the way very tiny changes at the start of a complex process can make big changes downstream. (The way that, for instance, the flapping of a butterfly's wings in Barcelona can eventually trigger a thunderstorm in Baltimore.)

But the Riemann has its biggest impact in the abstract world of mathematics.

For more than a century, mathematicians have dreamt up an estimated 300 theorems that assume Riemann is mathematical gospel. A proof would not only confirm those theorems but lead to many others as well.

"The hope is, it will give us the ultimate insight into whole numbers and prime numbers," says Peter Sarnak of Princeton University.

But all that is needed to disprove Riemann is a single example, somewhere in the far reaches of the number line, that doesn't fit the predicted pattern. If it is proven wrong, many mathematical ideas could collapse in the dust.

The notion scandalizes Bombieri. "I would be really astonished if it proved false," he says.

Surge in interest

Interest in the Riemann Hypothesis surged five years ago after Andrew Wiles of Princeton published the first valid proof of another long-standing math problem, Fermat's Last Theorem. Mathematicians had puzzled over that one for 356 years.

The feat made front-page news, though fewer than 20 people on Earth likely understood Wiles' tortuous 200-page mathematical proof, a logical argument that moves from assumption to conclusion. Some mathematicians think that Riemann, unlike Fermat, will have a simple proof -- an elegant solution that will cause them to wonder why they didn't see it all along.

Today, top-ranked mathematicians are working on Riemann at such schools as Princeton, Stanford and the College de France and at the tranquil, forest-ringed Institute for Advanced Study, where Albert Einstein and J. Robert Oppenheimer pondered the nature of the universe.

Bombieri is a senior professor at the institute. Atle Selberg, an expert on the Riemann, is professor emeritus. Nash, now 70, is also based here, although he has no office or formal title. With his illness in remission, Nash spends most of his days scribbling in his notebooks while sitting at a table in the institute's underground cafeteria.

Wiles is visiting here this semester and says he expects to spend at least some of his time working on Riemann.

During an afternoon tea in Fuld Hall's day room, Wiles sat on a leather couch and scrutinized a photo of a warped checkerboard pattern in a local weekly newspaper. He measured the sides of the squares with the tip of his ballpoint pen.

It might look as if Wiles is goofing off. But for mathematicians, it's a typical workday. Most science is increasingly pursued by teams of researchers using expensive gizmos. Mathematics, with its roots in ancient Mesopotamia, is mostly practiced by solitary folks who sit around and think, armed with little more than paper and pencil.

Says Bombieri: "Most of the thinking I do is to imagine, 'What if things were this way, or that?' 'What is missing?' Then I try to imagine building bridges" between what is known and what is unknown.

Ideas are the only things mathematicians create, so they guard them jealously.

Every day, Bombieri eats lunch in a section of the institute's cafeteria where members of the math department gather. They chat about politics, baseball, weather and, of course, their shared passion. But when it comes to math, Bombieri is very careful about what he says.

"I have to decide how much I want to share my ideas and how much I want to discuss with others," he says.

First algebra book at 8

The fourth child and only son of a banker in Milan, Italy, Bombieri read his first algebra book at age 8. He got serious about math, he says, as a 15-year-old in high school in Tuscany, when he picked up a book on number theory that introduced him to Riemann. He was hooked.

In 1974, while a 34-year-old professor at the University of Pisa, Bombieri won the Fields Medal, the mathematician's equivalent of a Nobel Prize. That same year he was offered a job at the institute.

Mathematicians are often stereotyped as absent-minded number nerds, "always immersed in his or her abstract thoughts," Bombieri complains. If so, he doesn't fit the mold.

With his powder-blue shirt open at the neck, khaki pants and running shoes, he might pass for an Italian film director at Cannes. Married with a grown daughter, he is a gourmet cook and a serious painter: He carries his paints and brushes with him whenever he travels.

Still, mathematics never seems far from his mind. In a recent painting, Bombieri, a one-time member of the Cambridge

University chess team, depicts a giant chessboard by a lake. He's placed the pieces to reflect a critical point in the historic match in which IBM's chess-playing computer, Deep Blue, beat Garry Kasparov.

Proposed proofs

Bombieri reads a lot of proposed proofs of Riemann, crafted by professional mathematicians and amateurs. (By tradition, many academic mathematicians review the work of amateurs.) He estimates that someone, somewhere, dreams up a new proof about once a week.

So far, of course, all are flawed. Still, Bombieri is optimistic. He points out that several mathematicians, including Alain Connes of the College de France, are working on new approaches. He hints that he's seen exciting, unpublished work.

He's so optimistic, he's wagered a bottle of fine champagne with a friend that a proof will surface by midnight Dec. 31, 2000.

But he's not nervous about the prospect of losing his bet.

Even unproved, the Riemann Hypothesis remains a towering intellectual landmark, the Mount Everest of math problems. Besides, Bombieri's friend lives in Princeton. They'll celebrate New Year's Eve together.

"Even if I pay for the champagne," he says, "I will get to taste it."

Pub Date: 9/30/98