Just Like The Movies

by Kate Melville

While Ernest Croot III, a graduate student in the mathematics department at the University of Georgia doesn't look much like Matt Damon in Good Will Hunting, he should. You might remember the film where Damon plays a university cleaner who turns out to be a mathematics genius? Well Ernest Croot did something very similar to the films story line when he announced his solution to a long-standing open question in mathematics posed by two famous mathematicians

The problem originated with Paul Erdos, circled the globe for fifty years working on mathematics problems, encouraging young mathematicians and posing questions that have led the way for researchers; and Ronald L. Graham, AT&T Labs chief scientist emeritus, distinguished professor of computer science at the University of California at San Diego, and treasurer of the National Academy of Sciences.

"Roughly speaking the problem asks whether, whenever you separate the whole numbers larger than 1 into several classes, there is a way of taking finitely many numbers from one of the classes such that the sum of their reciprocals is 1," said Dr. Andrew Granville, professor of mathematics. "For example, if 2, 3 and 6 all appeared in the same class, you would be through, since 1/2 1/3 1/6 = 1. So to defeat this, 2, 3 and 6 should not all appear in the same class, and likewise for every other way of writing 1 as a sum of reciprocals of finitely many whole numbers."

Croot, a graduate of Centre College, Ky., showed is that this cannot be defeated, there will always be some way to find one of these special sets in one of the classes. Superficially there do not, appear to be any practical applications of this result, but there may be some connections to deep questions in theoretical computer science.

During his life, Erdos offered prize money for his most stubborn problems, including \$500 for this particular problem. Before he died he signed some blank checks and left them in the care of Dr. Graham to present to solvers in the future. The check was presented to Croot after a lecture by John Selfridge, a famous mathematician in his own right and an old collaborator with Paul Erdos.

While Science A Go Go salutes Coot for this great bit of brain work, the unfortunate fact is that his \$500 prize is pretty small beer when you compare it to the fee Matt Damon must have received for playing a make believe mathematician (there's a moral in this somewhere).