Mathematical physicists have announced progress on a 150-year-old theorem for which the Clay Mathematical Institute is offering a million-dollar reward. Scientists have presented an operator that satisfies the Hilbert-Polya conjecture, which says that there is a differential operator whose eigenvalues exactly correspond to the nontrivial zeros of the Riemann zeta function. The article was published in the journal Physical Review Letters.

The Riemann Hypothesis is one of the Millennium Problems for which the American Clay Institute of Mathematics awards a million dollar prize. The Poincaré hypothesis (the Poincaré-Perelman theorem), which our compatriot proved, was included in this list. The Riemann Hypothesis, formulated in 1859, states that all non-trivial zeros of the Riemann zeta function (that is, the values ​​of the complex-valued argument that vanishes the function) lie on the line ½ + it, that is, their real part is equal to ½. The zeta function itself appears in many branches of mathematics, for example, in number theory, it is related to the number of primes less than a given one.

Function theory predicts that the set of non-trivial zeros of the zeta function should be similar to the set of eigenvalues ​​("solutions" for matrix equations) some other function from the class of differential operators, which are often used in physics. The idea of ​​the existence of a specific operator with such properties is called the Hilbert-Polya conjecture, although neither of them has published papers on this topic. “Since there are no publications by ‘authors’ on this topic, the formulation of the hypothesis changes depending on the interpretation,” explains one of the authors of the article, Dorje Brody from Brunel University in London. - However, two points must be met: a) one must find an operator whose eigenvalues ​​correspond to non-trivial zeros of the zeta function, and b) determine that the eigenvalues ​​are real numbers. The main goal of our work was point a). Further work is needed to prove part b).

Another important conjecture in this area is the idea of ​​Berry and Keating that if the desired operator exists, it will theoretically correspond to some quantum system with certain properties. “We determined the quantization conditions for the Berry-Keating Hamiltonian, thus proving the conjecture of their name,” adds Brodie. - It may be disappointing, but the resulting Hamiltonian does not seem to correspond to any physical system in an obvious way; at least we didn't find such a match."

The greatest difficulty is the proof of the validity of eigenvalues. The authors are optimistic about this, the article contains a supporting argument based on PT-symmetry. This idea from particle physics means that if all four-dimensional space-time directions are reversed, the system will look the same. Nature is generally not PT-symmetric, however, the resulting operator has this property. As shown in the article, if we prove the violation of this symmetry for the imaginary part of the operator, then all eigenvalues ​​will be real, thus completing the proof of the Riemann hypothesis.

Hello, habralyudi!

Today I would like to touch upon such a topic as the “millennium tasks”, which have been worrying the best minds of our planet for decades, and some even hundreds of years.

After proving the conjecture (now the theorem) of Poincaré by Grigory Perelman, the main question that interested many was: “ And what did he actually prove, explain on your fingers?» Taking the opportunity, I will try to explain on my fingers the other tasks of the millennium, or at least approach them from another side closer to reality.

Equality of classes P and NP

We all remember quadratic equations from school, which are solved through the discriminant. The solution to this problem is class P (P olynomial time)- for it, there is a fast (hereinafter, the word "fast" is meant as executing in polynomial time) solution algorithm, which is memorized.

There are also NP-tasks ( N on-deterministic P olynomial time), the found solution of which can be quickly checked using a certain algorithm. For example, check by brute-force computer. If we return to the decision quadratic equation, then we will see that in this example, the existing solution algorithm is checked as easily and quickly as it is solved. From this, a logical conclusion suggests itself that this task belongs to both one class and the second.

There are many such tasks, but the main question is whether all or not all tasks that can be easily and quickly checked can also be easily and quickly solved? Now, for some problems, no fast solution algorithm has been found, and it is not known whether such a solution exists at all.

On the Internet, I also met such an interesting and transparent wording:

Let's say that you, being in a large company, want to make sure that your friend is also there. If you are told that he is sitting in the corner, then a fraction of a second will be enough to, with a glance, make sure that the information is true. In the absence of this information, you will be forced to go around the entire room, looking at the guests.

AT this case the question is still the same, is there such an algorithm of actions, thanks to which, even without information about where a person is, find him as quickly as if knowing where he is.

This problem has great importance for the most various areas knowledge, but they have not been able to solve it for more than 40 years.

Hodge hypothesis

In reality, there are many simple and much more complex geometric objects. Obviously, the more complex the object, the more time-consuming it becomes to study. Now scientists have invented and are using with might and main an approach, the main idea of ​​which is to use simple "bricks" with already known properties that stick together and form its likeness, yes, a designer familiar to everyone since childhood. Knowing the properties of the "bricks", it becomes possible to approach the properties of the object itself.

Hodge's hypothesis in this case is connected with some properties of both "bricks" and objects.

Riemann hypothesis

Since school, we all know prime numbers that are divisible only by itself and by one. (2,3,5,7,11...) . Since ancient times, people have been trying to find a pattern in their placement, but luck has not smiled at anyone so far. As a result, scientists have applied their efforts to the prime number distribution function, which shows the number of primes less than or equal to a certain number. For example, for 4 - 2 prime numbers, for 10 - already 4 numbers. Riemann hypothesis just sets the properties of this distribution function.

Many statements about the computational complexity of some integer algorithms are proven under the assumption that this conjecture is true.

Yang-Mills theory

The equations of quantum physics describe the world elementary particles. Physicists Yang and Mills, having discovered the connection between geometry and elementary particle physics, wrote their own equations, combining the theories of electromagnetic, weak and strong interactions. At one time, the Yang-Mills theory was considered only as a mathematical refinement, not related to reality. However, later the theory began to receive experimental confirmation, but in general view it still remains unresolved.

On the basis of the Yang-Mills theory, the standard model of elementary particle physics was built within which the sensational Higgs boson was predicted and recently discovered.

Existence and smoothness of solutions of the Navier-Stokes equations

Fluid flow, air currents, turbulence. These and many other phenomena are described by equations known as Navier-Stokes equations. For some particular cases, solutions have already been found in which, as a rule, parts of the equations are discarded as not affecting the final result, but in general the solutions of these equations are unknown, and it is not even known how to solve them.

Birch-Swinnerton-Dyer hypothesis

For the equation x 2 + y 2 = z 2, Euclid once gave Full description solutions, but for more complex equations, finding solutions becomes extremely difficult, it is enough to recall the history of the proof of Fermat's famous theorem to see this.

This hypothesis is related to the description algebraic equations 3 degrees - the so-called elliptic curves and in fact is the only relatively simple in a general way rank calculation, one of the most important properties elliptic curves.

In proof Fermat's theorems elliptic curves have taken one of the most important places. And in cryptography, they form a whole section of the name itself, and some Russian digital signature standards are based on them.

Poincare conjecture

I think if not all, then most of you have definitely heard about it. Most often found, including in the central media, such a transcript as “ a rubber band stretched over a sphere can be smoothly pulled to a point, but a rubber band stretched over a donut cannot". In fact, this formulation is valid for the Thurston conjecture, which generalizes the Poincaré conjecture, and which Perelman actually proved.

A special case of the Poincare conjecture tells us that any three-dimensional manifold without boundary (the universe, for example) is like a three-dimensional sphere. And the general case translates this statement to objects of any dimension. It is worth noting that a donut, just like the universe is like a sphere, is like an ordinary coffee mug.

Conclusion

At present, mathematics is associated with scientists who have a strange appearance and talk about equally strange things. Many talk about her isolation from real world. Many people of both younger and quite conscious age say that mathematics is an unnecessary science, that after school / institute, it was not useful anywhere in life.

But in fact, this is not so - mathematics was created as a mechanism with which to describe our world, and in particular, many observable things. It is everywhere, in every home. As V.O. Klyuchevsky: “It’s not the flowers’ fault that the blind man doesn’t see them.”

Our world is far from being as simple as it seems, and mathematics, in accordance with this, is also becoming more complex, improving, providing more and more solid ground for a deeper understanding of the existing reality.

Is the Riemann Hypothesis proven?

A mathematician at Purdue University claims to have obtained a proof of the Riemann Hypothesis, often referred to as the greatest unsolved mathematical problem. Although the work of this mathematician still has to go through the peer review process.

This week the professor of mathematics at the School natural sciences Purdue, Edward Elliott Prize winner Louis de Branges published 23-page work with your proof. Usually mathematicians announce such achievements at conferences or in scientific journals. However, a prize of $1 million was awarded for proving the Riemann Hypothesis, so he decided to hurry with the publication. “I invite other mathematicians to check my calculations,” de Branges says in a prepared statement. “In time, I will submit my proof for official publication, but due to the circumstances, I feel the need to immediately publish my work on the Internet.”

The conjecture refers to the distribution of prime numbers. Prime numbers are only divisible by themselves and one. Among other tasks, prime numbers are used for encryption. Earlier this month, it was confirmed that the largest known prime number to date has been discovered, which is expressed by two to the power of 24036583 minus one and written as 7235733 in decimal digits.

Like solutions to many other mathematical problems, a proof of the Riemann hypothesis is unlikely to find an immediate commercial application, but in a decade its use is likely.

The origins of the conjecture date back to 1859, when the mathematician Bernhard Riemann proposed a theory about the distribution of prime numbers, but he died in 1866 before he could complete its proof. Since then, many have taken on the challenge. In particular, John Nash, a mathematician, Nobel laureate in economics, whose life story is the basis of the plot of the book and the movie, tried to solve it. A Beautiful Mind("Mind games"). In 2001, the Clay Mathematics Institute in Cambridge, Massachusetts, announced a $1 million prize for proving the conjecture.

De Branges is perhaps best known for solving another technical problem in mathematics: 20 years ago, he proved Bieberbach's theorem. Since then, the scientist has devoted himself almost entirely to testing the Riemann hypothesis.

Previous publications:

Discussion and comments

	nc Jun 10, 2004 12:21 PM
Respect to the man, at least for what he is trying to do.
	crest Jun 10, 2004 12:24 PM
Yes, the Nobel Prize in Mathematics is cool!
	torvic Jun 10, 2004 1:06 PM
"mathematician, Nobel Prize winner" [in economics]
	Yuri Abele Jun 10, 2004 1:17 PM
To Crest: John Nash is truly one of the greatest mathematicians of our time. Great not by the confusion of any mathematical calculations, but by the contribution that his work on game theory made to world economy. It practically turned the modern economy upside down. In a nutshell, he mathematically proved that it is more profitable for competitors, paradoxically, to cooperate rather than compete
	Maverick Jun 10, 2004 1:37 PM
2 torvic > John Nash, Nobel laureate mathematics This is the original. I almost fell off my chair! Apparently, the editors of zdnet haven't been paid for a long time. I'm not talking about the "gepotism" that shines in the annotation. Yes, no, the joke here is that the Nobel Prize in mathematics has long been a bearded historical anecdote.
	Qrot Jun 10, 2004 1:41 PM
> *Hypothesis* Riemann proven > proof of Riemann's *hypothesis* I remember our Russian teacher counted this as a double error. > ... by 1859, when the mathematician Bernhard Riemann proposed > theory... he died in 1966 What is he, a mountaineer? in the original "but he died in 1866" is there an editor besides the system administrator on call at all?
	Qrot Jun 10, 2004 1:44 PM
Nobel Prize-winning mathematician != Nobel laureate in mathematics. did you translate overhead?
	Maverick Jun 10, 2004 1:48 PM
As for the date of death, I did not pay attention. :-) Respect!
	Mikhail Elashkin- imhoelashkin.com Jun 10, 2004 2:07 PM
2 Qrot >overbrains translated? Oh, I see an attentive reader Goblin. Hello brother :)
	matros Jun 10, 2004 2:22 PM
2 Qrot: These are not overbrains, these are brainless. :)
	And Jun 10, 2004 3:22 PM
2 Yuri Abele. In my opinion, it is quite obvious that it is more profitable for competitors to cooperate rather than compete. In my opinion, such cooperation even has special names, such as "price collusion". And all sorts of antimonopoly authorities are trying to fight against such cooperation.
	Qrot Jun 10, 2004 4:23 PM
Mikhail Elashkin: salute to the comrade! :)
	Yuri Jun 10, 2004 6:32 PM
Well, noble nonsense was written here! Crap in almost every word. This is a special effort - and then you can’t think of it right away. The Riemann Hypothesis is, of course, related to the distribution of prime numbers (just like it is to many other interesting questions), but trying to explain its essence, starting with the concept of a prime number, is something special :-) And what does the discovery of another prime number have to do with the Riemann hypothesis, and even more so what commercial benefit could be derived from this proof, even if even after hundreds of years - this is generally a mystery to an inquisitive mind :-)
	bravomail Jun 10, 2004 7:09 PM
there is only one commercial benefit - the ease of breaking modern ciphers
	Yuri Jun 10, 2004 7:29 PM
> there is only one commercial benefit - the ease of breaking modern ciphers It is _absolutely_ independent not only of whether the Riemann Hypothesis is proved or not, but even on whether it is true at all.
	Ks Jun 10, 2004 8:57 PM
Generally speaking, the Riemann hypothesis concerns the zeros of the Riemann zeta function, and even if it is used in the theory of the distribution of prime numbers, then in a completely non-obvious way. Let's just say that Bertrand's postulate is proved using this very zeta function, but completely without this hypothesis.
	Nobody Jun 10, 2004 10:51 PM
Nobel to Lunix! Windows must die!
	done Jun 10, 2004 11:24 PM
2YuriВ what you sensible bring to our community??
	C3Man Jun 12, 2004 4:44 AM
APOLOGY FOR THE PROOF OF THE RIEMANN HYPOTHESIS?
	Alex Jun 13, 2004 6:15 PM
Previously, de Branges (that's the professor who claims to have proved the Riemann Hypothesis) proved a type theorem -- if a certain condition is true, then the Riemann Hypothesis is also true. Then it turned out that his condition was not true. In what hangs on the Internet, there is no proof of the Riemann hypothesis (and would you hang $ 1M on the Internet?), There is his apology to his colleagues that his proof may confuse his research plans, his path to proof and the fact that he would have done with 1M$. Hilbert once said that if he slept for 500 years and then woke up, the first thing he would ask is whether the Riemann hypothesis has been proven.
	Alex Jun 14, 2004 3:22 AM
Guilty, he actually posted the proof. Only not on 24 pages as it was initially reported, but on 124 pages. The man is 72 years old, and there is still gunpowder in the powder flasks and berries in the buttocks.
	Black ibm.* Jun 16, 2004 12:05 PM
In general, mathematics is good because in not "HOW a lot can be done" by a loner, sit and pick. You can't say the same about other sciences. EVEN theoretical physics where expensive experiments are not needed .. It is strongly connected with experimenters .. THAT THEORY PHYSICS only worked for experimenters (Landau YES a loner genius. BUT would he have achieved such a result if he hadn’t taken his Kapitsa?) .. well, except that Einstein stands apart . GOOD MAN.
	Nicholas Oct 13, 2006 2:34 PM
A few years ago I "proved" Fermat's Last Theorem. I was sooo happy, and then... I found a mistake! Is Mr. de Branges sure that he found a real proof? I-no!

Mathematical science. Work on them had a tremendous impact on the development of this area of ​​human knowledge. 100 years later, the Clay Mathematical Institute presented a list of 7 problems known as the Millennium Problems. Each of them was offered a prize of $1 million.

The only problem that appeared among both lists of puzzles that have been haunting scientists for more than one century was the Riemann hypothesis. She is still waiting for her decision.

Brief biographical note

Georg Friedrich Bernhard Riemann was born in 1826 in Hannover, in a large family of a poor pastor, and lived only 39 years. He managed to publish 10 works. However, already during his lifetime, Riemann was considered the successor of his teacher Johann Gauss. At the age of 25, the young scientist defended his dissertation "Fundamentals of the theory of functions of a complex variable." Later he formulated his hypothesis, which became famous.

prime numbers

Mathematics appeared when man learned to count. At the same time, the first ideas about numbers arose, which they later tried to classify. It has been observed that some of them have common properties. In particular, among natural numbers, i.e., those that were used in counting (numbering) or designating the number of objects, a group of those that were divided only by one and by themselves was singled out. They are called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his Elements. On the this moment their search continues. In particular, the largest of the already known is the number 2 74 207 281 - 1.

Euler formula

Along with the concept of the infinity of the set of primes, Euclid also defined the second theorem on the only possible decomposition into prime factors. According to it, any positive integer is the product of only one set of prime numbers. In 1737, the great German mathematician Leonhard Euler expressed Euclid's first infinity theorem in the form of the formula below.

It is called the zeta function, where s is a constant and p takes on all prime values. Euclid's statement about the uniqueness of the expansion directly followed from it.

Riemann zeta function

Euler's formula, on closer inspection, is absolutely amazing, as it defines the relationship between primes and integers. After all, on its left side, infinitely many expressions that depend only on prime numbers are multiplied, and on the right side there is a sum associated with all positive integers.

Riemann went further than Euler. In order to find the key to the problem of the distribution of numbers, he proposed to define a formula for both real and complex variables. It was she who subsequently received the name of the Riemann zeta function. In 1859, the scientist published an article entitled "On the number of prime numbers that do not exceed a given value", where he summarized all his ideas.

Riemann suggested using the Euler series, which converges for any real s>1. If the same formula is used for complex s, then the series will converge for any value of this variable with a real part greater than 1. Riemann applied the analytic continuation procedure, extending the definition of zeta(s) to all complex numbers, but "thrown out" the unit. It was excluded because for s = 1 the zeta function increases to infinity.

practical meaning

A natural question arises: what is interesting and important about the zeta function, which is the key to Riemann's work on the null hypothesis? As you know, at the moment no simple pattern has been identified that would describe the distribution of prime numbers among natural numbers. Riemann was able to discover that the number pi(x) of primes that did not exceed x is expressed in terms of the distribution of non-trivial zeros of the zeta function. Moreover, the Riemann hypothesis is necessary condition to prove time estimates for the operation of some cryptographic algorithms.

Riemann hypothesis

One of the first formulations of this mathematical problem, which has not been proven to this day, sounds like this: non-trivial 0 zeta functions are complex numbers with real part equal to ½. In other words, they are located on the line Re s = ½.

There is also a generalized Riemann hypothesis, which is the same statement, but for generalizations of zeta functions, which are usually called Dirichlet L-functions (see photo below).

In the formula χ(n) is some numerical character (modulo k).

The Riemannian assertion is considered the so-called null hypothesis, as it has been tested for consistency with existing sample data.

As Riemann argued

The remark of the German mathematician was initially formulated rather casually. The fact is that at that time the scientist was going to prove the theorem on the distribution of prime numbers, and in this context, this hypothesis did not have much meaning. However, its role in solving many other issues is enormous. That is why Riemann's assumption is currently recognized by many scientists as the most important of the unproven mathematical problems.

As already mentioned, the full Riemann hypothesis is not needed to prove the distribution theorem, and it is sufficient to justify logically that the real part of any non-trivial zero of the zeta function is in the interval from 0 to 1. From this property it follows that the sum over all 0-th The zeta functions that appear in the exact formula above are a finite constant. For large values ​​of x, it may be lost altogether. The only member of the formula that remains the same even for very large x is x itself. The remaining complex terms vanish asymptotically in comparison with it. So the weighted sum tends to x. This circumstance can be considered a confirmation of the truth of the theorem on the distribution of prime numbers. Thus, the zeros of the Riemann zeta function have a special role. It lies in the fact that the values ​​cannot make a significant contribution to the expansion formula.

Followers of Riemann

The tragic death from tuberculosis did not allow this scientist to bring his program to its logical end. However, he received baton W-F. de la Vallée Poussin and Jacques Hadamard. Independently of each other, they deduced a theorem on the distribution of prime numbers. Hadamard and Poussin succeeded in proving that all nontrivial 0 zeta functions are within the critical band.

Thanks to the work of these scientists, a new direction in mathematics appeared - the analytic theory of numbers. Later, several more primitive proofs of the theorem that Riemann was working on were obtained by other researchers. In particular, Pal Erdős and Atle Selberg even discovered a very complex logical chain confirming it, which did not require the use of complex analysis. However, by this point, several important theorems had already been proved by means of Riemann's idea, including the approximation of many functions of number theory. Concerning new job Erdős and Atle Selberg had practically no effect on anything.

One of the simplest and most beautiful proofs of the problem was found in 1980 by Donald Newman. It was based on the famous Cauchy theorem.

Does the Riemannian Hypothesis threaten the foundations of modern cryptography?

Data encryption arose along with the advent of hieroglyphs, more precisely, they themselves can be considered the first codes. At the moment, there is a whole area of ​​digital cryptography, which is developing

Prime and "semi-prime" numbers, that is, those that are divisible only by 2 other numbers from the same class, underlie the system with public key, known as RSA. She has widest application. In particular, it is used when generating an electronic signature. Speaking in terms accessible to dummies, the Riemann hypothesis asserts the existence of a system in the distribution of prime numbers. Thus, the strength of cryptographic keys, on which the security of online transactions in the field of e-commerce depends, is significantly reduced.

Other unresolved mathematical problems

It is worth finishing the article by devoting a few words to other millennium tasks. These include:

• Equality of classes P and NP. The problem is formulated as follows: if a positive answer to a particular question is checked in polynomial time, is it true that the answer to this question itself can be found quickly?
• Hodge hypothesis. In simple words it can be formulated as follows: for some types of projective algebraic varieties (spaces), Hodge cycles are combinations of objects that have a geometric interpretation, i.e., algebraic cycles.
• The Poincaré hypothesis. This is the only Millennium Challenge that has been proven so far. According to it, any 3-dimensional object that has the specific properties of a 3-dimensional sphere must be a sphere up to deformation.
• Statement of the quantum theory of Yang-Mills. It is required to prove that quantum theory, put forward by these scientists for the space R 4 , exists and has a 0th mass defect for any simple compact gauge group G.
• Birch-Swinnerton-Dyer hypothesis. This is another issue related to cryptography. It concerns elliptic curves.
• The problem of the existence and smoothness of solutions of the Navier-Stokes equations.

Now you know the Riemann hypothesis. In simple terms, we have formulated some of the other Millennium Challenges. That they will be solved or it will be proved that they have no solution is a matter of time. And it is unlikely that this will have to wait too long, since mathematics is increasingly using the computing capabilities of computers. However, not everything is subject to technology, and first of all, intuition and creativity are required to solve scientific problems.

Editorial response

Michael Francis Atiyah, a professor at Oxford, Cambridge and Edinburgh Universities and winner of almost a dozen prestigious awards in mathematics, presented a proof of the Riemann Hypothesis, one of the seven Millennium Problems, which describes how the prime numbers are located on the number line.

Atiyah's proof is short, taking up five pages, together with the introduction and bibliography. The scientist claims that he found a solution to the hypothesis by analyzing the problems associated with the fine structure constant, and used the Todd function as a tool. If the scientific community considers the proof correct, then the Briton will receive $ 1 million for it from the Clay Mathematics Institute (Clay Mathematics Institute, Cambridge, Massachusetts).

Other scientists are also vying for the prize. In 2015, he announced the solution of the Riemann hypothesis Professor of Mathematics Opeyemi Enoch from Nigeria, and in 2016 presented his proof of the hypothesis Russian mathematician Igor Turkanov. According to representatives of the Institute of Mathematics, in order for an achievement to be recorded, it must be published in an authoritative international magazine followed by confirmation of the evidence by the scientific community.

What is the essence of the hypothesis?

The hypothesis was formulated back in 1859 by the German mathematician Bernhard Riemann. He defined a formula, the so-called zeta function, for the number of primes up to a given limit. The scientist found that there is no pattern that would describe how often prime numbers appear in the number series, while he found that the number of prime numbers that do not exceed x, is expressed in terms of the distribution of the so-called "non-trivial zeros" of the zeta function.

Riemann was confident in the correctness of the derived formula, but he could not establish on what simple statement this distribution completely depends. As a result, he put forward the hypothesis that all non-trivial zeros of the zeta function have a real part equal to ½ and lie on the vertical line Re=0.5 of the complex plane.

The proof or refutation of the Riemann hypothesis is very important for the theory of the distribution of prime numbers, says PhD student of the Faculty of Mathematics of the Higher School of Economics Alexander Kalmynin. “The Riemann Hypothesis is a statement that is equivalent to some formula for the number of primes not exceeding a given number x. A hypothesis, for example, allows you to quickly and with great accuracy calculate the number of prime numbers that do not exceed, for example, 10 billion. This is not the only value of the hypothesis, because it also has a number of rather far-reaching generalizations, which are known as the generalized Riemann hypothesis , the extended Riemann hypothesis, and the grand Riemann hypothesis. They have more greater value for different branches of mathematics, but first of all, the importance of the hypothesis is determined by the theory of prime numbers,” says Kalmynin.

According to the expert, with the help of a hypothesis, it is possible to solve a number of classical problems of number theory: Gauss problems on quadratic fields (the problem of the tenth discriminant), Euler's problems on convenient numbers, Vinogradov's conjecture on quadratic non-residues, etc. In modern mathematics, this hypothesis is used to prove statements about prime numbers. “We immediately assume that some strong hypothesis like the Riemann hypothesis is true, and see what happens. When we succeed, we ask ourselves: can we prove it without assuming a hypothesis? And, although such a statement is still beyond what we can achieve, it works like a beacon. Due to the fact that there is such a hypothesis, we can see where we are going,” says Kalmynin.

The proof of the hypothesis can also affect the improvement information technologies, since the encryption and encoding processes today depend on the effectiveness of different algorithms. “If we take two simple large numbers of forty digits and multiply, then we will get a large eighty-digit number. If we set the task to factorize this number, then this will be a very complex computational task, on the basis of which many information security issues are built. All of them consist in creating different algorithms that are tied to the complexities of this kind, ”Kalmynin says.