The oldest math problem in history has found an answer

Mathematics has always made people dream. Since Antiquity, the Egyptians in particular have used numbers to perfect their world. But although full of great scholars, the Egyptian world could remain unanswered in the face of certain problems. This was particularly the case with the problem described as “ the oldest ever » by the mathematician Carl Pomerance, who notably works at the prestigious Dartmouth College, member of the American Ivy League.

With a very simple title, this problem known since ancient Egypt has still not found an answer. But we are surely closer to the solution than we have ever been. Indeed, a mathematician from the University of Oxford, named Thomas Bloom, has just provided part of the answer.

A story of fractions

In order to understand the full prowess of Bloom’s work, one must first understand this problem. It concerns unit fractions with a denominator, such as 1/2 or 1/4. These fractions were practically the only ones used in Egypt at the time apart from 2/3 and probably 3/4. All other fractions were broken down into sums of unit fractions such as, for example, 2/7 = 1/4 + 1/28.

These very particular fractions regained a certain interest in the 1970s when two mathematicians (Paul Erdős and Ronald Graham) conjectured that “Any set that samples a sufficiently large positive proportion of the integers must contain a subset of the integers whose inverses s‘add up to get the sum of 1. As an example we can cite the subset {2,3,6} of the set {1,2,8,3,6} since 1/2 + 1/3 + 1/6 = 1.

A well-known enigma in the world of mathematics, which has long been a real headache for the most eminent researchers, such as Andrew Granville, of the University of Montreal who ensures “ that‘no sane person‘mind can never answer “.

And despite the warnings of the British researcher, his colleague Thomas Bloom has embarked on this thousand-year-old calculation. He began by taking up the work of Ernie Croot, who studied the question for him for a long time in the late 1990s, then in the early 2000s. This researcher said he had solved the “colored” version of this problem.

The color buckets, a first answer

Ernie Croot arranges the whole numbers in buckets of different colors in a random fashion. Erdős and Graham predicted that in this case no matter how many seals are used, there will be at least one that will contain the desired subset. Croot employed modern methods from harmonic analysis to demonstrate this conjecture.

Since Croot’s article was already impressive, it still did not give the full answer to the question posed by Erdős and Graham. It still did not indicate the seal in which the subset sought was.

Bloom followed the method used by Croot which employs a type of integral called the exponential sum. It was necessary to estimate with the sign of this sum the existence of at least one solution of the problem. Bloom adopted Croot’s strategy but faced the problem Croot wanted to avoid: whole numbers that have large prime factors and don’t give reducing fractions.

Bloom therefore undertook an impressive number of calculations, and although in certain situations the exponential sum seemed to be negative (it is almost impossible to find the exact result of this kind of calculation), other choices of numbers made it possible to rebalance the balance.

A huge addition, to unlock a thousand-year-old secret

He finally decided to proceed almost by trial and error. As it added inverses of whole numbers together, it kept some results aside (like 1/3 for example). Once he had rolled 1/3 three times, he would roll 1, as the riddle asks.

Although this research did not put an end to the enigma, the new method employed by Bloom did not fail to react within the scientific community. For Izabella Laba, researcher at the Canadian University of British Columbia “ vs‘is an exceptional result ».

But Bloom’s discoveries did not end the mystery entirely. On the contrary, the latter is now more open than ever. The big question ultimately comes down to finding the exceptions that exist within Bloom’s rule. Flaws where the inverses of integers do not equal 1.

In conclusion, the Erdős-Graham conjecture still holds many secrets for us, and it will be very interesting to see in which areas of the real world this research will have an impact.

A concrete application to hope for?

The study of this kind of part of mathematics makes it possible in particular to develop the knowledge of scientists in very concrete subjects such as computer science or coding. Despite everything, it will surely be necessary to wait years, even decades, for concrete applications to be found by taking into account this problem, and the answers provided by Thomas Bloom in his research.

By then, other mathematicians will surely have unearthed other great problems, centuries old, just to rack their brains again, and, in passing, to advance science and knowledge of the world.

Article written in collaboration with Alena Fonade, doctoral student and professor of mathematics