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By Phil Williams
Ernest S. Croot III, a graduate student in mathematics, surprised mathematicians at the University of Illinois Number Theory Conference in Champaign, Ill., recently by announcing his solution to a long-standing open question in mathematics posed by two famous mathematicians.
The problem originated with Paul Erdos, who toured the world for half a century working on mathematics problems, encouraging young mathematicians and posing interesting research questions, 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,” says Andrew Granville, the David C. Barrow Professor of Mathematics at UGA. “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. 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, showed that this proposition cannot be defeated--there will always be some way to find one of these special sets in one of the classes. Although there do not, on the surface, appear to be any practical applications of this result, it may well be that there are connections with deep questions in theoretical computer science.
In the early part of this summer, at a conference in Budapest celebrating the life and work of Paul Erdos, Ronald Graham was a plenary speaker and discussed some of his favorite memories of Paul Erdos, including this particular question that both of them had worked on fruitlessly for many years.
When he was alive, Erdos offered prize money for his most stubborn problems, including $500 for the solution to this problem.
Before he died he signed some blank checks and left them in the care of Graham to present to future problem solvers.
Initially, Croot shared his insights with Granville, his adviser, and with former UGA Research Professor Carl Pomerance, now senior researcher at Lucent Technologies’ Bell Labs. Discussing matters with them (by e-mail with Pomerance), he polished his argument and got ready to announce his work at the Illinois meeting. Pomerance tipped off Graham as to what was about to happen, and Graham made out one of the Erdos blank checks to Croot for $500.
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