Number theorists look for hidden structures. When confronted with a numerical pattern that seems unavoidable, they try hard, and often fail, to come up with situations in which a given pattern cannot appear.
One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that go all the way back to ancient Egypt.
Carl Pomerance said that it might be the oldest problem ever.
The question is about fractions with a 1 in their numerator. The ancient Egyptians had only one type of fractions in their number system. They could only express more complicated fractions with the exception of a single symbol.
The interest in such sums got a boost in the 1970s when Ronald Graham and Paul Erd asked how hard it would be to engineer sets of whole numbers that don't contain a subset. The set 2, 3, 6, 9, 13 fails this test because it contains the subset 2, 3, 6.
It could be 20 percent or 1 percent or 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- 888-609- If the initial set is able to satisfy that simple condition of sampling enough whole numbers, then the subset would have to exist.
Andrew Granville of the University of Montreal thought that the question was impossible to answer.
He was asked to present a 20-year-old paper to a reading group at Oxford.
The coloring version of the problem had been solved by that paper. The whole numbers are sorted by color into different buckets, with some going in the blue bucket and others in the red one. No matter how many different buckets are used, at least one bucket has to contain a subset of whole numbers.
Croot introduced new methods from a branch of math that is related to calculus. The Annals of Mathematics is the top journal in the field.
It requires creativity, ingenuity, and a lot of technical strength, according to the University of Georgia.
The paper was impressive, but it couldn't answer the density version of the question. Croot took advantage of the convenience of the bucket-sorting formula, but not the density one.
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