The New Answer is found in the ‘Oldest Problem’ of Math

The numerators always looking for the secret house. And compared to an unknown number, they try their best, try hard – and often fail – to think of situations for which a given reason cannot be identified.

One of the last results that demonstrates the stability of such methods, Thomas Bloom of Oxford University, answers a question with roots that go all the way back to ancient Egypt.

“That’s probably the oldest problem,” said Carl Pomerance of Dartmouth College.

The question applies to fractions involving 1 in their number, such as 1⁄2, 1⁄7, or 1⁄122. These “fragments” were very important to the ancient Egyptians because they were the only fragments in their numerical system. With a difference of one symbol for 2⁄3, they can express hard fractions (e.g. 3⁄4) as compound fractions (1⁄2 + 1⁄4).

Today’s interest in such numbers increased in 1970, when Paul Erdes and Ronald Graham asked how difficult it was to engineer sets of whole numbers that did not have a segment. that the reciprocals are added to 1. For example, the setting {2, 3, 6, 9, 13} fails in this test: There is the part set {2, 3, 6}, whose reciprocals are the part fractions 1⁄2, 1⁄3, and 1⁄6 – in 1.

More precisely, Erdős and Graham believe that each group representing a significant and positive proportion of the overall scores – which can be 20 percent or 1 percent or 0.001 percent – should have a Part where reciprocals are added to 1. If the initial setting is satisfactory. Such a simple method of collecting whole numbers (known as “positive density”), then if its members are chosen to be difficult to find in that segment, the subset will remain the same.

“I just thought this was an impossible question that none of their right mind could work on,” said Andrew Granville of the University of Montréal. “I don’t know of any tool that can attack him.”

Bloom’s connection to Erdős and Graham’s question grew out of a home study: Last September, he was asked to donate a 20 -year paper to a reading group in Oxford.

That paper, by a mathematician named Ernie Croot, solved what he called the color scheme of the Erdős-Graham problem. There, the whole numbers are separated into different buckets chosen by colors: one goes to the blue bucket, the other to the red, and so on. Erdős and Graham predicted that because of the many different buckets used in this separation, one bucket should contain a fraction of the total numbers whose shares are equal to 1.

Croot introduced powerful innovations from harmonic analysis – a branch of numerical mathematics – to confirm the Erdős -Graham prediction. His paper was published in History of mathematicsthe top journal on the field.

“Croot’s argument is fun to read,” said Giorgis Petridis of the University of Georgia. “It takes thinking, ingenuity, and a lot of technical strength.”

As good as Croot’s paper is, however, it cannot respond to the strong influence of Erdős-Graham’s theory. Due to the convenience Croot used the material in the comparison to the bucket, but not in the density.

A mathematical scroll called the Rhind Papyrus, which dates back to 1650 BCE, shows how the ancient Egyptians presented correct numbers as fractions.Photo: Alamy

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