Mathematics gives some of the most dramatic examples of the glacial but inexorable advance of the human intellect. For example, in 1994 British mathematician Andrew Wiles revealed his proof of Fermat's last theorem. Wiles used esoteric mathematical tools like elliptical equations, unheard of when Pierre de Fermat first scribbled the problem in the margin of a book 350 years earlier. "I have a truly marvelous proof of this proposition, which this margin is too narrow to contain," Fermat had claimed. He died before his note was discovered, his proof apparently unwritten. For seven years, Wiles toiled secretly in his attic; only his wife knew that he was unraveling one of math's most enduring mysteries.

Just this August, Russian mathematician Grigori Perelman won a Fields Medal, math's equivalent of the Nobel, for proving the Poincaré conjecture (renamed the Poincaré theorem as a result). Like Wiles, Perelman carried out years of work in private; his proof, which he posted online, shook the math community. The conjecture, which states that the simplest three-dimensional shapes are all spheres in disguise, had resisted proof for more than 100 years, and Perelman's approach expanded the work of other mathematicians. But in a stunning development, Perelman refused the C$15,000 Fields Medal, saying that if his proof held, he needed no other recognition.

This raises the question: What problems remain? Plenty. Every area of mathematics has unsolved mysteries, and every so often mathematical listmakers target a few. In 1900 German mathematician David Hilbert enumerated 23 problems that would guide mathematics in the 20th century. At least three of Hilbert's problems remain unresolved. In 2000 the Clay Mathematics Institute in Cambridge, Massachusetts, identified seven Millennium Prize problems (the Poincaré was one). The solutions each carry a million-dollar prize. The Poincaré is the only one of the seven problems solved. Here is a sampling of some of the great problems lurking in math's netherworld. Each promises its solver a place in the mathematical pantheon. Three are attached to the million-dollar Clay bounty.

RIEMANN HYPOTHESIS

Prime numbers are positive integers divisible only by themselves and 1 (such as 2, 3, 5, and 7) and seem to occur at random. For centuries, mathematicians have sought unsuccessfully to find a pattern to their distribution. The Riemann hypothesis offers mathematicians a glimmer of hope. Often referred to as the holy grail of unsolved problems, it implies that while prime numbers display unpredictable patterns, there is an underlying orderliness determined by the zeros of the Riemann zeta function.

This function is also the cornerstone of volumes of literature. "If we proved the Riemann hypothesis, thousands of papers might have to be thrown in the wastebasket," says mathematician Dan Goldston. "Hundreds of mathematicians would have to change." Studies have verified the hypothesis for more than 1.5 billion cases, but verification is not tantamount to proof. Is anyone close? "I think we can predict earthquakes better than we can predict when it will be solved," says Jim Carlson, president of the Clay institute.

TWIN PRIME CONJECTURE

Goldston, of San Jose State University, has spent a quarter of a century studying the patterns of the primes. Unlike Wiles and Perelman, Goldston says he likes to work out in the open. "I report as I go and try to get people interested," he says. Three years ago, he and his colleague Cem Yildirim presented a paper that, if correct, would have verified this conjecture, one of the golden oldies in the field of number theory. More than 2,000 years ago, in a proof often referred to as elegant, the Greek mathematician Euclid proved that an infinite number of primes exist. His conclusion inspired another hypothesis. Some pairs of prime numbers differ by two—for example, 5 and 7 or 41 and 43. On the high end of the number line, these pairs of primes, called twins, are few and far between; despite their scarcity, mathematicians believe an infinite number of these pairs probably exists. The pursuit of the twin prime conjecture has led many a mathematician down blind alleys. Goldston's work with Yildirim contained a fatal flaw, and only part of the analysis could be repaired. But one mathematician's fumble can become another's inspiration. British mathematician Ben Green and 2006 Fields Medal–winner Terence Tao found, buried within the manuscript, a tool they needed to make their breakthrough on progressions of primes. The experience has also pointed Goldston in new directions.

THE NAVIER-STOKES EQUATIONS

Eddies spin behind a speeding boat, smoke billows from an erupting volcano, and an airplane lurches through "empty" air. All encounter turbulence, which appears only when it acts on something else. "Big whorls have little whorls that feed on their velocity, and little whorls have smaller whorls and so on," says physicist L. F. Richardson. The random motions of turbulence are found everywhere in life, and yet they are fantastically difficult to model. Mathematicians and physicists believe that an explanation for turbulence might be buried in the tangle of Greek letters known as the Navier-Stokes equations. The first of these equations extends Newton's second law of motion to fluids that might experience turbulence; the second equation ensures that the fluid is incompressible (its density doesn't change with fluctuations in pressure). From blood flow to weather patterns, the equations prove invaluable in modeling turbulent environments; even so, physically meaningful solutions are nearly impossible to come by.

THE TRAVELING SALESMAN

Think of this as the "Willy Loman" problem or, in light of the greeting-card blizzard of the holiday season, the mailman's dilemma.

A mailman (or Mr. Loman) is assigned to hit every house in a given city. If we know how long it takes to go from house to house, is it possible to determine the most time-efficient route? Intrepid explorers often look at all possible routes, then select the shortest. For a city with just two houses, the task is easy—there are two routes. For a city with four houses, 24 possible routes exist (4 x 3 x 2 x 1 = 24). Of these 24, it should be easy to determine the best route. But for a city with 20 houses, our mailman would have to examine roughly 2,432,902,008,000,000,000 (2.4 billion billion) different routes and would be better off just going mailbox to mailbox haphazardly. Calculating a route for a city of any real size—say, one with tens of thousands of houses—would stymie the most sophisticated computer in the world. Can anyone write a program that can find the most efficient solution? An answer to this question, a classic in the field of computational complexity theory, will net its finder a million dollars from the Clay.