For example, in 1994 British mathematician Andrew Wiles reveal his proof of Fermats last theorem. Wiles used esoteric mathematical tols 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 to narrow to contain.” Fermat had claimed. He died before his note was discovered, his proof apparently unwritten. For seven long 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 equivelent of the Nobel, for proving Poincare conjecture (renamed the Poincare theorem as a result.) Like Wiles, Perelma carried out years of work in private; his proof, which he posted on line, 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 real stunning development, Perelman refused the $15,000 Fields Medal, saying that if his proof held, he needed no other recognition.

* * * * * * *


                                                   WHAT PROBLEMS REMAIN?


Every area of mathematics has unsolved mysteries, and every so often mathematical listmakers target a few.

In 1900 German mathematician David Hillbert enumerated 23 problems that guide mathematicians in the coming 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 Poincare was one.) THE SOLUTIONS EACH CARRY A MILLION DOLLAR PRIZE. The Poincare is the only one of the seven problems solved.

Here is a sampling of the some of the great problems lurking in math’s nether world. Each promises its solver a place in the mathematical pantheon. Three of them are attached to the million-dollar Clay bvounty.


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 now, 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 as while prime numbers display unpredictable patterns , there is an underlying orderliness determined by the zeros of the Reemann zeta function.

This function is also the cornerstone of volumes of literature. “If we proved the Reimann hypothesis , thousand of papers might have to thrown in the wastebasket” says 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, presidne to the Clay institute.


Goldston, of San Jose State University, has spent a quarter of a century the pattern 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 the conjecture, one oif 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 —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 exist. The pursuit of the twin prime conjecture has led many a mathematician down blind alleys. Gold’s work with Yildirim contained a fatal flaw and only part of the analysis could be repaired. But one mathematicians fumble can become another’s inspiration. British mathematicians Ben Green and 2006 Fields Medal-winner Terence Tao found, buried within the manuscript, a tool they needed to make their breakthrough on progression of primed. The experience has also pointed Goldston in new directions.


Eddies spin behind a fast 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.


Think of this as the “Willie 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 simple—there are two routes. Now, for a city with four houses, 24 possible routs 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 rioutes and would be better off just going mailbox to mailbox haphazardly. Calculating a rioute for a city of any real size ----say, one with tens of thousands of houses of hiuses----would stymie the most sophisticated computer in the world today.

Can anyone write a program that can find the most efficient solution?

An answer to this quwstion , a classic in the field of computational complexity theory, will net its finder a cool million dollars from the Clay.

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