Math’s ‘oldest problem’ gets a new answer

number theorists Always looking for hidden structures. And when faced with numerical patterns that seem unavoidable, they test their often-failing mettle by working hard to devise situations in which a given pattern cannot appear.

one of latest results To show the elasticity of these patterns Thomas Bloom Doctors from Oxford University answer questions with roots that date back to ancient Egypt.

“It may be the oldest problem.” Carl Pomerance of Dartmouth College.

The problem is with fractions with 1 in the numerator, such as 1⁄2, 1⁄7, or 1⁄122. This “unit fraction” was of particular importance to the ancient Egyptians, because they were the only type of fraction included in the number system. Except for the single sign for 2⁄3, more complex fractions (such as 3⁄4) could be expressed as the sum of unit fractions (1⁄2 + 1⁄4).

Modern interest in these sums was boosted in the 1970s when Paul Erdős and Ronald Graham asked how difficult it is to engineer a set of integers that does not contain a subset whose reciprocal adds to one. For example, the set {2, 3, 6, 9, 13} will fail this test. It contains the subset {2, 3, 6} whose reciprocals are unit fractions 1⁄2, 1⁄3, 1⁄6. One.

More precisely, Erdős and Graham conjectured that every set that samples a sufficiently large positive fraction of an integer (which could be 20% or 1% or 0.001%) must contain a subset whose reciprocal is added to 1. The simple condition of sampling enough integers (known as having a “positive density”), then the subset must exist nonetheless, even if its members are deliberately chosen to make that subset difficult to find.

“I thought this was an impossible question that no sane person could ask,” he said. Andrew Granville of the University of Montreal. “I didn’t see any obvious tools to attack it.”

It was from homework that Bloom got involved in Erdossi and Graham’s questions. Last September, he was asked to present a 20-year-old paper to a reading group in Oxford.

The thesis was written by a mathematician called Ernie Crute, solved the so-called colored version of the Erdős-Graham problem. From there, the integers are randomly sorted into different color-coded buckets. Some go into the blue bucket, some go into the red bucket, and so on. Erdős and Graham predicted that regardless of the number of buckets used for this sort, at least one bucket must contain a subset of integers whose reciprocal sum is 1.

Croot introduced a powerful new method in harmonic analysis, a branch of mathematics closely related to calculus, to confirm Erdős-Graham predictions. his thesis is published in math yearbookThe best journals in the field.

“Kroot’s argument is a pleasure to read.” Georges Petridis of the University of Georgia. “It takes creativity, ingenuity, a lot of technical strength.”

However, while Croot’s paper was impressive, it could not answer the dense version of the Erdős-Graham conjecture. This was due to the convenience of Croot, which can be used in bucket sort formulas but not in density.

A mathematical scroll known as the lind papyrus, dating back to around 1650 BC, shows how the ancient Egyptians expressed rational numbers as sums of unit Alamy

Math’s ‘oldest problem’ gets a new answer

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