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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Zhang Hezhi

The expression of integers as the sum of fractions can be traced back to mathematical problems in ancient Egypt more than 3000 years ago, and ancient Egyptian fractions, which are closely related to them, still inspire mathematicians' curiosity. In the 1980s, the famous mathematician Eldespar guessed that any set of integers "large enough" could be finally combined by summing their reciprocals, 1 but he did not prove his conjecture. Recently, this conjecture that has lasted for 40 years has been solved.

Ancient Egyptian fraction archaeology found that ancient Egyptians had mastered a considerable degree of mathematical knowledge thousands of years ago. Despite the vicissitudes of life, some documents have been handed down, the most famous of which are Rhind Mathematical Papyrus and Moscow Mathematical Papyrus. Dozens of questions and answers are recorded on these papyrus, one of which is: how to make individuals slice bread evenly? In terms of the mathematics involved, this problem seems too obvious to us today: it only needs to be divided into pieces. But archaeologists found that in ancient Egypt, there was no such number! In the existing literature, except for and, all the scores appear in the form of. This may be because it is easier to divide a loaf of bread evenly when dividing bread. The ancient Egyptians seemed to need the help of a score table when calculating, which also occupied a certain space in the above-mentioned ancient books. And the answer to the above question given by the ancient Egyptians is

In other words, each piece of bread is equally divided into pieces, and each of the remaining pieces is equally divided into pieces, and each of the remaining pieces is equally divided into pieces, and each of the remaining pieces is divided equally, and finally each person takes a piece. Today, we call the score of shape as the score of ancient Egypt. Of course, every score can be divided into the sum of ancient Egyptian fractions, because

A

Of course, this is not difficult. This shows that the scores of the ancient Egyptians, though complex, are the same as the scores we use today: no matter how evenly divided a loaf of bread gets a rational number. But if you ask the other way around, it is much more difficult: if you choose a set of integers that are different and larger than each other, can you use their partial reciprocal to make a loaf of bread? In other words, for a set, whether there is a subset that satisfies

If it is limited, then there must be a fixed answer, just exhaustive. For example, at that time, we could borrow the example above.

But it's just luck; if so, it won't make it at all. After all, the total is smaller than the sum. You must say: it's because it's too small. If it were a little bigger, wouldn't it be?

So this time we take all the primes. There are infinitely many primes, but a subset of primes cannot construct the above equation: if there are different primes that satisfy

that

This shows that divisible, which is different from their prime contradiction, shows that the above equation does not exist. Why is there such a phenomenon? Isn't the set of prime numbers big enough?

In set theory, the size is generally compared by mapping: if there is an one-to-one corresponding relationship between the sum of two sets, it is called equipotential. If it is equipotential with a subset of. But for the sum of infinite sets, it is possible to be equal to its own subset. It can be proved that, in the sense of equipotential, the set of natural numbers is the smallest infinite set, and all the infinite subsets it contains are equipotential, including the set of primes. But in terms of the problems encountered above, the prime set is obviously not large enough. Then we need to define a new concept of size.

The rational number of prizes recorded in Reinder's papyrus is expressed as scores and picture sources: Alamy Stock Photo natural density, so what is the "big" we need? In fact, the core demand is: whether we are odd or even, plain or close, as long as it contains just a few numbers we need. This actually coincides with the view of probability: grab a handful of numbers at hand, and if the proportion is large enough, you are more likely to catch numbers that meet the requirements.

How to calculate this probability? The number of numbers that are not greater than in our definition is, then the probability of being randomly selected in the whole is

It's called natural density. However, this limit is not always convergent, so the upper limit is generally used.

Called the upper density, and the lower limit.

It's called lower density. When the sequence does not converge, it tends to swing up and down, and the upper and lower limits are the upper and lower bounds of the swing. The concept of natural density is very consistent with our intuition, for example, the natural density of even and odd sets is exactly half. What about the density of prime numbers? According to the famous prime theorem, when it is big enough

That is to say, the natural density of primes is. This is in line with what we expected: the set of primes is indeed too small. So how much does the natural density of a set have to hold at least a subset of the reciprocal sum? In this regard, two mathematicians boldly put forward a guess.

In an article [1] in 1980, the Hungarian mathematician Erd's P á l (Paul Erd's in English) and the American mathematician Ronald Graham raised the problem of decomposing it into ancient Egyptian scores, which can be divided into two versions using the symbols above:

If we divide the larger integers into finite parts (or, in the words of mathematicians, dye the numbers into different colors), does one have to contain the finite subset we want?

If, does it include what we want?

Of the shares given in the previous question, at least one share has a natural density greater than zero. So if you prove the latter question, you will prove the previous question. Such second consecutive questions are not made out of nothing. In number theory, every proposition involving coloring corresponds to a proposition involving natural density: although it is proved that the previous question cannot be deduced from the latter question, the method used can lead us further.

It is not easy to find a new way to analyze number theory, but it is not easy to prove the first question. This kind of number theory research involving addition is generally called additive number theory or stacked number theory. This branch believes that many mathematics enthusiasts have heard of it, because Hua Luogeng, Chen Jingrun and others, representatives of the Chinese school of analytic number theory, have made great contributions in this regard. Hua Luogeng also wrote the book "the Theory of Prime numbers"; the well-known Goldbach conjecture also belongs to this field.

For most of the problems of additive number theory, elementary and even algebraic methods are often used to fail one after another, and only analytic number theory is left. The so-called analytical number theory is to transform the problem of number theory into the estimation of a function or integral (the so-called analysis is analysis, that is, the discipline involving limits, calculus and its derivatives). This is also the reason why additive number theory is too abstract for the layman: it is obviously a number theory problem, but the process of proof is full of integrals and estimates.

The same is true of Eldesh and Geliheng's conjecture. Eldesh, who died in 1996, was unmarried and had no children; he was left with only 1525 papers, making him the mathematician with the largest number of papers published. Until the end of his life, he did not see his conjecture confirmed. A few years later, in 2003, Ernest S. Croot III's paper [2] came out, proving the first question of conjecture. It is worth mentioning that as early as 2000, Klute proved this conclusion in his doctoral thesis. Klute introduces a powerful method of harmonic analysis, which is both elegant and skillful. This makes the academic circles look forward to the new star.

The so-called harmonic, which is generally translated as harmonics in physics, comes from Fourier's amazing discovery that many periodic functions can be decomposed into infinite sums of trigonometric functions, that is, Fourier series. What is more amazing is that Fourier series and Fourier integral can be used to estimate some functions in number theory, which closely links harmonic analysis with analytical number theory. Since then, the two disciplines have supported each other and developed rapidly under the abstract tide of modern mathematics in the 20th century.

But faced with the second question, Klute's ingenious method failed. No matter how he fiddles with the existing tools, there is no way to make more progress. Since then, Klute has turned to other issues, and our conjecture has stagnated for two decades. By 2020, GE Liheng died of illness, and he was not able to see the solution of the conjecture.

The turnaround took place in 2021. One day in September, Thomas Bloom (Thomas Bloom), a postdoctoral fellow at the University of Oxford, was assigned the task of explaining to their discussion group Klute's paper two decades ago. In the process of preparation, Bloom was suddenly inspired-Klute's method did not come to an end! He set about the work at once.

Although there has been no progress in conjecture in the past two decades, cutting-edge mathematics such as harmonic analysis has not stopped. Now, Bloom has more tools in his hand. He improved Klute's method by using more advanced combinatorial / analytic number theory techniques, and finally completed the proof in a few months.

Bloom proved that the conclusion was even stronger than the original guess-as long as it was fine. In other words, the sequence does not have to converge, it only needs to oscillate infinitely near a positive number. This is a very excellent result, although it has not been officially published on the preprint website, it has been recognized by many mathematicians.

So now, this conjecture that has spanned 40 years has finally come to an end. But there is no end to math; apart from the dazzling skills, we still have a few questions: which sets cannot be summed up? As mentioned above, the prime set does not meet the requirements. However, not all sets with natural density do not meet the requirements.

Where will more new questions lead us? Let's wait and see.

reference

[1] Erd's, P. and Graham, R. L.: Old and new problems and results in combinatorial number theory. Enseign. Math. 30-44 (1980)

Croot, Ernest S., III (2003). "On a coloring conjecture about unit fractions". Annals of Mathematics. 157 (2): 545-556. ArXiv:math.NT/0311421. Doi:10.4007/annals.2003.157.545. MR 1973054.

Cepelewicz, Jordana (2022-03-09). "Math's' Oldest Problem Ever' Gets a New Answer". Quanta Magazine. Retrieved 2022-03-09.

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