MILLION-DOLLAR CONTEST CREATES A MILLION HEADACHES

by David Appell

Motivated by million-dollar prizes for solving some of the world's most difficult scientific puzzles, amateur mathematicians and physicists around the world are inundating academic journals with what they hope are winning entries. But instead of being thankful for the attention, journal editors wish they'd just go away.

"They're really coming out of the woodwork," said David Goss, a professor of mathematics at Ohio State University and editor-in-chief of the Journal of Number Theory. "At times, I am almost getting more crank stuff than legitimate stuff."

Editors at the Annals of Mathematics report a similar increase in crank submissions.
Serving as bait are the seven "Millennium Prizes" offered last May by the Clay Mathematics Institute of Cambridge, MA. Worth $1 million each, they will be awarded to the first person who publishes an accepted, peer-reviewed paper on some of the most important outstanding problems in mathematics and physics.

But academic math journals, often run by professors in their spare time, weren't set up to referee math contests. And the flood of crank entries is so overwhelming that people with a legitimate shot at solving the problems say they aren't getting a fair hearing.

Charles Francis, for instance, thinks he solved one of the problems: proof that certain equations used to describe elementary particles, called the Yang-Mills equations, actually have solutions when the particles have mass (which, of course, most do), and why quarks-subatomic particles-are never seen outside the confines of their protons or neutrons. Though he worked in the field while obtaining his Ph.D. from the University of London, the independent software developer for Clef Digital Systems in Aberystwyth, Wales, can't get any respect from the likes of the International Journal of Theoretical Physics.

Most say "they cannot afford to provide proper reviews," Francis said in an e-mail correspondence. "I am not employed by an academic institution, and I can only suppose that is the reason I have not been able get my work peer reviewed by any journal to which I submit."

Yet, finding solutions to these problems is much more than a contest. One math problem, a proof or disproof of the Riemann Hypothesis, is the most important unsolved problem in mathematics, said Goss of the Journal of Number Theory. A speculation of the brilliant German mathematician Georg Friedrich Bernard Riemann in 1859, it has important implications for many areas of mathematics, including the way prime numbers (those divisible only by one and themselves) are distributed throughout the number system, and in cryptography. Almost every mathematician has cut a tooth on it at some point in their training, and most believe it contains deep insights into mathematical truth.

"Right now, when we tackle problems without knowing the truth of the Riemann Hypothesis, it's as if we have a screwdriver," Princeton University mathematician Peter Sarnak told New Scientist magazine. "But when we have it, it'll be more like a bulldozer."

Goss said he recently received a one-page proof of the Hypothesis, among others. "I look to see if it has a snowball's chance," he said. "If it doesn't, I reject it." (It didn't.)

Most professional mathematicians and physicists say they are not motivated by multi-million-dollar prizes and believe the guarantee of earning a place in history is enough of a prize for solving any of the famous conundrums. They frequently cite as an example Princeton mathematician Andrew Wiles' fame gained in 1994 for solving Fermat's Last Theorem.

But mathematics isn't exactly a lucrative business for most people, and some of these award-seekers could certainly use the money.

Take unemployed physicist Matti Pitkanen of Finland, who recently submitted a paper claiming a proof to the Riemann Hypothesis, only to back away from it a few weeks later. Pitkanen, a self-described "scientific dissident" working in a 215-square-foot apartment, said his original motivation came elsewhere, but that the prize has crossed his mind.

"I have four children, and I could buy a nice little house and see them during weekends," he said. Like some others, he believes that ideas from physics, especially quantum theory and chaos theory, hold the keys to settling Riemann's nagging question, and he continues to work hard on the problem.

"No serious mathematician needs a monetary prize to work on one of these problems," said number theorist Daniel Bump of Stanford University, who focuses on the Riemann Hypothesis. "One could argue that if the goal of the prize is to generate a solution of one of these problems, the prize money would be better spent in supporting research proposals in the targeted areas."

The Clay Mathematical Institute itself has been besieged with proposed solutions to the Millennium problems, even though the rules specifically says it will not review any papers until two years after they're published by a peer-reviewed academic journal.

"We've had probably 600 or more people send things to us," said Arthur Jaffe, president of the Clay Mathematics Institute and a professor of mathematics at Harvard University. "The level of people out there who are attracted to these problems is extraordinary, and yet almost all the things that come in the mail are really not serious."

The most popular problems (characterized by Jaffe as anything where people think they can just write down some numbers and that's the solution) are Riemann's Hypothesis and the Birch and Swinnerton-Dyer Conjecture. The latter asks for the number of whole number solutions (integers such as 3, 16, and 258) to quadratic equations that are akin to the Pythagorean Theorem in high school.

The other Millennium problems are deceptively simple to state. One asks if the equations that describe the flow of fluids, the Navier-Stokes equations, have solutions where the fluid flows smoothly. (How physicists have managed to use this equation for nearly 200 years without knowing if it actually has solutions is, perhaps, an implicit problem.) An active group of mathematical physicists has been addressing this problem for years, and cranks seem scared away by its more complicated look.

"What we're trying to do at the Clay Institute in naming these prizes is to really focus attention on mathematics in the general public," said Jaffe, "and I think that's been very successful."

Mathematics has a long tradition of posed problem sets, though monetary rewards are a relatively recent introduction. Asking "Who of us would not be glad to lift the veil behind which the future lies hidden?" the towering German mathematician David Hilbert posed a famous list of 23 problems at the 1900 International Congress of Mathematicians in Paris. Arguably the most influential speech ever given about mathematics, Hilbert's problems extended over all fields of mathematics and served as a roadmap for 20th century mathematics. Three of them are not yet fully solved, including the Riemann Hypothesis.

Among Hilbert's originals, the Goldbach Conjecture also has a million-dollar bounty on its head, set last March by the British publishing house Faber and Faber. Posed in 1742, the problem is to prove that every even integer greater than 2 can be represented as the sum of two primes, for example, 8 = 3+5 or 58 = 17+41. Computer scientists have verified the conjecture up to 400 trillion. Unlike the Clay Prizes, which have no time limit, this prize offer expires in 2002.

The Electronic Freedom Foundation (EFF) has a group of four prizes, the Cooperative Computing Awards, totaling $550,000. The first asked for the first million-digit prime number, and its $50,000 was awarded last April to Nayan Hajratwala of Michigan. Hajratwala found a two-million-digit prime number, the largest yet known, using the collective power of tens of thousands of computers on the Entropia.com network. The remaining prizes ask for still larger primes (with the last reward for one of a billion digits), and have served to motivate important new work.

"It's stimulated people cooperating together to work on very difficult problems," said Landon Curt Noll, a member of the awards advisory panel and himself a past holder of seven prime number-related world records. "Together we can pool our resources to accomplish things that as individuals we cannot."

One of the Millennium Prizes is directly related to computer science, the "P vs. NP problem." It asks whether questions exist whose answer can be quickly checked by, say, a computer, but which require a much longer time to solve from scratch. (It's relatively easy to calculate that 3607 X 3803 = 13,717,421, but far harder to find the two numbers which, when multiplied together, give the latter number.)

Last year this question was shown to be related to the computer game Minesweeper, which comes pre-installed on PCs using Microsoft's Windows.

If you can prove there's a logically slick way to win the game on an arbitrarily large Minesweeper board, the million bucks are yours. Most experts believe no such way exists-disprove that and the prize is yours as well. But without a simple-looking formula to gnaw on, this problem attracts less amateur solutions than some of the others.

"Cranks come in all sizes and shapes but what they have in common is their obsessiveness," said Underwood Dudley, a mathematics professor at DePauw University and by his own reckoning the world's only expert on mathematical cranks. Dudley attracts their papers like a magnet, and was once sued by one for calling him a crank. (After two appeals and an order to pay all legal fees, the crank stopped.)

"They aren't crazy, usually, and you can't tell them apart from other people by looking," he says. "They are odd only in one direction."

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David Appell is a freelance writer based in Gilford, NH. Who Wants To Be a Math Millionaire?, March 27, 2001, Boston Globe. © David Appell.