Math’s ‘Oldest Problem Ever’ Gets a New Answer

Number theorists are at all times on the lookout for hidden construction. And when confronted by a numerical sample that appears unavoidable, they check its mettle, making an attempt onerous—and sometimes failing—to plan conditions through which a given sample can not seem.

One of the newest outcomes to show the resilience of such patterns, by Thomas Bloom of the University of Oxford, solutions a query with roots that reach all the way in which again to historical Egypt.

“It might be the oldest problem ever,” mentioned Carl Pomerance of Dartmouth College.

The query entails fractions that function a 1 of their numerator, like 1⁄2, 1⁄7, or 1⁄122. These “unit fractions” had been particularly necessary to the traditional Egyptians as a result of they had been the one forms of fractions their quantity system contained. With the exception of a single image for two⁄3, they may solely specific extra difficult fractions (like 3⁄4) as sums of unit fractions (1⁄2 + 1⁄4).

The modern-day curiosity in such sums received a lift within the Nineteen Seventies, when Paul Erdős and Ronald Graham requested how onerous it may be to engineer units of complete numbers that don’t comprise a subset whose reciprocals add to 1. For occasion, the set {2, 3, 6, 9, 13} fails this check: It accommodates the subset {2, 3, 6}, whose reciprocals are the unit fractions 1⁄2, 1⁄3, and 1⁄6—which sum to 1.

More precisely, Erdős and Graham conjectured that any set that samples some sufficiently massive, constructive proportion of the entire numbers—it might be 20 % or 1 % or 0.001 %—should comprise a subset whose reciprocals add to 1. If the preliminary set satisfies that easy situation of sampling sufficient complete numbers (often known as having “positive density”), then even when its members had been intentionally chosen to make it tough to seek out that subset, the subset would nonetheless need to exist.

“I just thought this was an impossible question that no one in their right mind could possibly ever do,” mentioned Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”

Bloom’s involvement with Erdős and Graham’s query grew out of a homework task: Last September, he was requested to current a 20-year-old paper to a studying group at Oxford.

That paper, by a mathematician named Ernie Croot, had solved the so-called coloring model of the Erdős-Graham drawback. There, the entire numbers are sorted at random into completely different buckets designated by colours: Some go within the blue bucket, others within the pink one, and so forth. Erdős and Graham predicted that regardless of what number of completely different buckets get used on this sorting, at the very least one bucket has to comprise a subset of complete numbers whose reciprocals sum to 1.

Croot launched highly effective new strategies from harmonic evaluation—a department of math intently associated to calculus—to verify the Erdős-Graham prediction. His paper was printed within the Annals of Mathematics, the highest journal within the discipline.

“Croot’s argument is a joy to read,” mentioned Giorgis Petridis of the University of Georgia. “It requires creativity, ingenuity, and a lot of technical strength.”

Yet as spectacular as Croot’s paper was, it couldn’t reply the density model of the Erdős-Graham conjecture. This was resulting from a comfort Croot took benefit of that’s obtainable within the bucket-sorting formulation, however not within the density one.

The mathematical scroll often known as the Rhind Papyrus, which dates again to round 1650 BCE, reveals how the traditional Egyptians represented rational numbers as sums of unit fractions.Photograph: Alamy

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