So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in its essence and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n > 2. Everything seems to be simple and clear, but the best mathematicians and simple amateurs fought over searching for a solution for more than three and a half centuries.

Why is she so famous? Now let's find out...



Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with the 5th grade of secondary school, but the proof is far from even every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved for so long. 2. What does it consist of?

Let's start with Pythagorean pants The wording is really simple - at first glance. As we know from childhood, "Pythagorean pants are equal on all sides." The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs.

In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triples satisfying the equation x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They probably tried to look for triples and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.


That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²

Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.

Or 5, 12, 13: 25 + 144 = 169. Great.

Well, and so on. What if we take a similar equation x³+y³=z³ ? Maybe there are such numbers too?




And so on (Fig. 1).

Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.

It is more difficult to prove the absence: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give a solution). And that's it, the opponent is defeated. How to prove absence?

To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.

In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2):


And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:





But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically studied the general equation x n+yn=zn . And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! All that remains is his remark in Diophantus' Arithmetic: "I have found a truly amazing proof of this proposition, but the margins here are too narrow to accommodate it."

Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.

After Fermat, great minds such as Leonhard Euler worked on finding the proof (in 1770 he proposed a solution for n = 3),

Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat's Last Theorem, but only in 1993 did mathematicians see and believe that the three-century saga of finding a proof of Fermat's last theorem was almost over.

It is easy to show that it suffices to prove Fermat's theorem only for prime n: 3, 5, 7, 11, 13, 17, … For composite n, the proof remains valid. But there are infinitely many prime numbers...

In 1825, using the method of Sophie Germain, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.


Finally, the German mathematician Ernst Kummer showed in a brilliant study that the methods of mathematics in the 19th century cannot prove the theorem in general terms. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.

In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.

He soon died of natural causes. The heirs were pretty surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...

Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:


Dear (s). . . . . . . .

Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau











In 1963, Paul Cohen, drawing on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems, the continuum hypothesis. What if Fermat's Last Theorem is also unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups of programmers and mathematicians proved Fermat's Last Theorem for all values ​​of n up to 500, then up to 1,000, and later up to 10,000.

In the 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians claimed that Fermat's Last Theorem was true for all values ​​of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.




In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.

Nevertheless, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole building could collapse at any moment.

In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proved that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proved. But for thirty years it was not possible to prove the Taniyama–Shimura conjecture, and there were less and less hopes for success.

In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.

Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama–Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat’s Last Theorem is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.

In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.







While the hype continued in the press, serious work began to verify the evidence. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.

It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.

“...half a minute after the start of the festive dinner on the occasion of her birthday, I gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?






This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.

A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!

Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ... - » Tasks of Humanity

TASKS OF MATHEMATICS UNSOLVED BY HUMANITY

Hilbert problems

The 23 most important problems in mathematics were presented by the greatest German mathematician David Hilbert at the Second International Congress of Mathematicians in Paris in 1990. Then these problems (covering the foundations of mathematics, algebra, number theory, geometry, topology, algebraic geometry, Lie groups, real and complex analysis, differential equations, mathematical physics, calculus of variations and probability theory) were not solved. So far 16 problems have been solved out of 23. Another 2 are not correct mathematical problems (one is formulated too vaguely to understand whether it is solved or not, the other, far from being solved, is physical, not mathematical) Of the remaining 5 problems, two are not solved in any way, and three are solved only for some cases

Landau problems

Until now, there are many open questions related to prime numbers (a prime number is a number that has only two divisors: one and the number itself). The most important questions were listed Edmund Landau at the Fifth International Mathematical Congress:

Landau's first problem (Goldbach's problem): is it true that every even number greater than two can be represented as the sum of two primes, and every odd number greater than 5 can be represented as the sum of three primes?

Landau's second problem: Is the set infinite? "simple twins"- prime numbers, the difference between which is equal to 2?
Landau's third problem(Legendre's conjecture): is it true that for any natural number n between and there is always a prime number?
Landau's fourth problem: Is the set of prime numbers of the form , where n is a natural number, infinite?

Millennium targets (Millennium Prize Problems

These are seven math problems, h and the solution to each of which the Clay Institute offered a prize of 1,000,000 US dollars. Bringing these seven problems to the attention of mathematicians, the Clay Institute compared them with D. Hilbert's 23 problems, which had a great influence on the mathematics of the twentieth century. Of Hilbert's 23 problems, most have already been solved, and only one, the Riemann hypothesis, has been included in the list of millennium problems. As of December 2012, only one of the seven millennium problems (the Poincaré hypothesis) has been solved. The prize for her solution was awarded to the Russian mathematician Grigory Perelman, who refused it.

Here is a list of these seven tasks:

No. 1. Equality of classes P and NP

If a positive answer to a question is possible fast check (using some supporting information called a certificate) whether the answer itself (together with the certificate) to this question is true fast find? Problems of the first type belong to the NP class, and of the second type to the class P. The problem of the equality of these classes is one of the most important problems in the theory of algorithms.

No. 2. Hodge hypothesis

An important problem in algebraic geometry. The conjecture describes cohomology classes on complex projective varieties realized by algebraic subvarieties.

Number 3. The Poincaré hypothesis (proved by G.Ya. Perelman)

It is considered the most famous topology problem. More simply, it states that any 3D "object" that has some properties of a three-dimensional sphere (for example, every loop inside it must be contractible) must be a sphere up to deformation. The prize for the proof of the Poincaré conjecture was awarded to the Russian mathematician G.Ya.

No. 4. Riemann hypothesis

The conjecture states that all non-trivial (that is, having a non-zero imaginary part) zeros of the Riemann zeta function have a real part of 1/2. The Riemann hypothesis was the eighth in Hilbert's list of problems.

No. 5. Yang-Mills theory

A task from the field of elementary particle physics. It is required to prove that for any simple compact gauge group G the quantum Yang-Mills theory for a four-dimensional space exists and has a nonzero mass defect. This statement is consistent with experimental data and numerical simulations, but it has not yet been proven.

No. 6. Existence and smoothness of solutions of the Navier-Stokes equations

The Navier-Stokes equations describe the motion of a viscous fluid. One of the most important problems in hydrodynamics.

No. 7. Birch-Swinnerton-Dyer hypothesis

The hypothesis is related to the equations of elliptic curves and the set of their rational solutions.

"All I know is that I don't know anything, but others don't know that either"
(Socrates, ancient Greek philosopher)

NOBODY is given to own the universal mind and know EVERYTHING. Nevertheless, most scientists, and even those who simply love to think and explore, always have a desire to learn more, to solve mysteries. But are there still unsolved topics in humanity? After all, it seems that everything is already clear and you just need to apply the knowledge gained over the centuries?

Do NOT despair! There are still unsolved problems from the field of mathematics, logic, which in 2000 the experts of the Clay Mathematical Institute in Cambridge (Massachusetts, USA) combined into a list of the so-called 7 mysteries of the Millennium (Millennium Prize Problems). These problems concern scientists all over the world. From then to this day, anyone can claim to have found a solution to one of the problems, prove a hypothesis, and receive an award from Boston billionaire Landon Clay (after whom the institute is named). He has already allocated $7 million for this purpose. By the way, Today, one of the problems has already been solved.

So, are you ready to learn about math riddles?

Navier-Stokes equations (formulated in 1822)

Field: hydroaerodynamics

The equations for turbulent, air, and fluid flows are known as the Navier-Stokes equations. If, for example, you float on a lake on something, then waves will inevitably arise around you. This also applies to airspace: when flying in an airplane, turbulent flows will also form in the air.
These equations just produce description of the processes of motion of a viscous fluid and are the core problem of all hydrodynamics. For some particular cases, solutions have already been found in which parts of the equations are discarded as having no effect on the final result, but solutions to these equations have not been found in general terms.
It is necessary to find a solution to the equations and identify smooth functions.

Riemann hypothesis (formulated in 1859)

Field: number theory

It is known that the distribution of prime numbers (which are divisible only by themselves and by one: 2,3,5,7,11…) among all natural numbers does not follow any regularity.
The German mathematician Riemann thought about this problem, who made his assumption, theoretically concerning the properties of the existing sequence of prime numbers. The so-called paired prime numbers have long been known - twin prime numbers, the difference between which is equal to 2, for example, 11 and 13, 29 and 31, 59 and 61. Sometimes they form whole clusters, for example, 101, 103, 107, 109 and 113 .
If such accumulations are found and a certain algorithm is derived, this will lead to a revolutionary change in our knowledge in the field of encryption and to an unprecedented breakthrough in the field of Internet security.

Poincare problem (formulated in 1904. Solved in 2002.)

Field: topology or geometry of multidimensional spaces

The essence of the problem lies in the topology and lies in the fact that if you stretch a rubber band, for example, on an apple (sphere), then it will be theoretically possible to compress it to a point, slowly moving the tape without taking it off the surface. However, if the same tape is pulled around a donut (torus), then it is not possible to compress the tape without breaking the tape or breaking the donut itself. Those. the entire surface of a sphere is simply connected, while that of a torus is not. The task was to prove that only the sphere is simply connected.

Representative of the Leningrad Geometric School Grigory Yakovlevich Perelman is the recipient of the Clay Institute of Mathematics Millennium Prize (2010) for solving the Poincaré problem. He refused the famous Fildes Prize.

Hodge hypothesis (formulated in 1941)

Field: algebraic geometry

In reality, there are many simple and much more complex geometric objects. The more complex the object, the more difficult it is to study it. Now scientists have invented and are using with might and main an approach based on the use of parts of one whole ("bricks") to study this object, as an example - a designer. Knowing the properties of the "bricks", it becomes possible to approach the properties of the object itself. The Hodge hypothesis in this case is connected with some properties of both "bricks" and objects.
This is a very serious problem in algebraic geometry: to find exact ways and methods to analyze complex objects with the help of simple "bricks".

Yang-Mills equations (formulated in 1954)

Field: geometry and quantum physics

Physicists Yang and Mills describe the world of elementary particles. They, having discovered the connection between geometry and elementary particle physics, wrote their own equations in the field of quantum physics. Thereby a way was found to unify the theories of electromagnetic, weak and strong interactions.
At the level of microparticles, an “unpleasant” effect arises: if several fields act on a particle at once, their combined effect can no longer be decomposed into the action of each of them one by one. This is due to the fact that in this theory, not only particles of matter are attracted to each other, but also the field lines themselves.
Although the Yang-Mills equations are accepted by all physicists of the world, the theory concerning the prediction of the mass of elementary particles has not been experimentally proven.

Birch and Swinnerton-Dyer hypothesis (formulated in 1960)

Field: algebra and number theory

Hypothesis related to the equations of elliptic curves and the set of their rational solutions. In the proof of Fermat's theorem, elliptic curves occupied 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.
The problem is that you need to describe ALL solutions in integers x, y, z of algebraic equations, that is, equations in several variables with integer coefficients.

Cook's problem (formulated in 1971)

Field: mathematical logic and cybernetics

It is also called "Equality of classes P and NP", and it is one of the most important problems in the theory of algorithms, logic and computer science.
Can the process of checking the correctness of the solution of a problem last longer than the time spent on the solution of this problem itself(regardless of the verification algorithm)?
The solution of the same problem, sometimes, takes a different amount of time, if you change the conditions and algorithms. For example: in a large company you are looking for a friend. If you know that he is sitting in a corner or at a table, then it will take you a split second to see him. But if you do not know exactly where the object is, then spend more time looking for it, bypassing all the guests.
The main question is: can all or not all problems that can be easily and quickly checked be also easily and quickly solved?

Mathematics, as it may seem to many, is not so far from reality. It is the mechanism by which our world and many phenomena can be described. Math is everywhere. And V.O. was right. Klyuchevsky, who said: "It's not the flowers' fault that the blind can't see them".

In conclusion….

One of the most popular theorems in mathematics - Fermat's Last Theorem: an + bn = cn - could not be proved for 358 years! And only in 1994 the Briton Andrew Wiles was able to give her a solution.

Unsolvable problems are 7 most interesting mathematical problems. Each of them was proposed at one time by well-known scientists, as a rule, in the form of hypotheses. For many decades, mathematicians all over the world have been racking their brains over their solution. Those who succeed will be rewarded with a million US dollars offered by the Clay Institute.

Clay Institute

This name is a private non-profit organization headquartered in Cambridge, Massachusetts. It was founded in 1998 by Harvard mathematician A. Jeffey and businessman L. Clay. The aim of the Institute is to popularize and develop mathematical knowledge. To achieve this, the organization gives awards to scientists and sponsors promising research.

At the beginning of the 21st century, the Clay Mathematical Institute offered a prize to those who solve problems that are known as the most difficult unsolvable problems, calling their list Millennium Prize Problems. From the "Hilbert List" it included only the Riemann hypothesis.

Millennium Challenges

The Clay Institute list originally included:

• the Hodge cycle hypothesis;
• equations of quantum theory Yang-Mills;
• the Poincaré hypothesis;
• the problem of equality of classes P and NP;
• the Riemann hypothesis;
• on the existence and smoothness of its solutions;
• Birch-Swinnerton-Dyer problem.

These open mathematical problems are of great interest because they can have many practical implementations.

What did Grigory Perelman prove

In 1900, the famous philosopher Henri Poincaré suggested that any simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere. Its proof in the general case was not found for a century. Only in 2002-2003, the St. Petersburg mathematician G. Perelman published a number of articles with a solution to the Poincaré problem. They had the effect of an exploding bomb. In 2010, the Poincaré hypothesis was excluded from the list of "Unsolved Problems" of the Clay Institute, and Perelman himself was offered to receive a considerable remuneration due to him, which the latter refused without explaining the reasons for his decision.

The most understandable explanation of what the Russian mathematician managed to prove can be given by imagining that a rubber disk is pulled onto a donut (torus), and then they try to pull the edges of its circumference into one point. Obviously this is not possible. Another thing, if you make this experiment with a ball. In this case, a seemingly three-dimensional sphere, resulting from a disk, the circumference of which was pulled to a point by a hypothetical cord, will be three-dimensional in the understanding of an ordinary person, but two-dimensional from the point of view of mathematics.

Poincaré suggested that a three-dimensional sphere is the only three-dimensional "object" whose surface can be contracted to a single point, and Perelman was able to prove this. Thus, the list of "Unsolvable problems" today consists of 6 problems.

Yang-Mills theory

This mathematical problem was proposed by its authors in 1954. The scientific formulation of the theory is as follows: for any simple compact gauge group, the quantum spatial theory created by Yang and Mills exists, and at the same time has a zero mass defect.

Speaking in a language understandable to an ordinary person, the interactions between natural objects (particles, bodies, waves, etc.) are divided into 4 types: electromagnetic, gravitational, weak and strong. For many years, physicists have been trying to create a general field theory. It should become a tool for explaining all these interactions. Yang-Mills theory is a mathematical language with which it became possible to describe 3 of the 4 main forces of nature. It does not apply to gravity. Therefore, it cannot be considered that Yang and Mills succeeded in creating a field theory.

In addition, the nonlinearity of the proposed equations makes them extremely difficult to solve. For small coupling constants, they can be approximately solved in the form of a series of perturbation theory. However, it is not yet clear how these equations can be solved with strong coupling.

Navier-Stokes equations

These expressions describe processes such as air flows, fluid flow, and turbulence. For some special cases, analytical solutions of the Navier-Stokes equation have already been found, but so far no one has succeeded in doing this for the general one. At the same time, numerical simulations for specific values ​​of speed, density, pressure, time, and so on can achieve excellent results. It remains to be hoped that someone will be able to apply the Navier-Stokes equations in the opposite direction, that is, calculate the parameters with their help, or prove that there is no solution method.

Birch-Swinnerton-Dyer problem

The category of "Unsolved Problems" also includes the hypothesis proposed by English scientists from the University of Cambridge. Even 2300 years ago, the ancient Greek scientist Euclid gave a complete description of the solutions to the equation x2 + y2 = z2.

If for each of the prime numbers to count the number of points on the curve modulo it, you get an infinite set of integers. If you specifically “glue” it into 1 function of a complex variable, then you get the Hasse-Weyl zeta function for a third-order curve, denoted by the letter L. It contains information about the behavior modulo all prime numbers at once.

Brian Burch and Peter Swinnerton-Dyer conjectured about elliptic curves. According to it, the structure and number of the set of its rational solutions are related to the behavior of the L-function at the identity. The currently unproven Birch-Swinnerton-Dyer conjecture depends on the description of 3rd degree algebraic equations and is the only relatively simple general way to calculate the rank of elliptic curves.

To understand the practical importance of this task, it is enough to say that in modern cryptography a whole class of asymmetric systems is based on elliptic curves, and domestic digital signature standards are based on their application.

Equality of classes p and np

If the rest of the Millennium Challenges are purely mathematical, then this one is related to the actual theory of algorithms. The problem concerning the equality of the classes p and np, also known as the Cooke-Levin problem, can be formulated in understandable language as follows. Suppose that a positive answer to a certain question can be checked quickly enough, i.e., in polynomial time (PT). Then is the statement correct that the answer to it can be found fairly quickly? Even simpler it sounds like this: is it really not more difficult to check the solution of the problem than to find it? If the equality of the classes p and np is ever proved, then all selection problems can be solved for PV. At the moment, many experts doubt the truth of this statement, although they cannot prove the opposite.

Riemann hypothesis

Until 1859, no pattern was identified that would describe how prime numbers are distributed among natural numbers. Perhaps this was due to the fact that science dealt with other issues. However, by the middle of the 19th century, the situation had changed, and they became one of the most relevant that mathematics began to deal with.

The Riemann Hypothesis, which appeared during this period, is the assumption that there is a certain pattern in the distribution of prime numbers.

Today, many modern scientists believe that if it is proven, then many of the fundamental principles of modern cryptography, which form the basis of a significant part of the mechanisms of electronic commerce, will have to be revised.

According to the Riemann hypothesis, the nature of the distribution of prime numbers may differ significantly from what is currently assumed. The fact is that so far no system has been discovered in the distribution of prime numbers. For example, there is the problem of "twins", the difference between which is 2. These numbers are 11 and 13, 29. Other prime numbers form clusters. These are 101, 103, 107, etc. Scientists have long suspected that such clusters exist among very large prime numbers. If they are found, then the stability of modern crypto keys will be in question.

Hodge Cycle Hypothesis

This hitherto unsolved problem was formulated in 1941. Hodge's hypothesis suggests the possibility of approximating the shape of any object by "gluing" together simple bodies of higher dimensions. This method has been known and successfully used for a long time. However, it is not known to what extent the simplification can be made.

Now you know what unsolvable problems exist at the moment. They are the subject of research by thousands of scientists around the world. It remains to be hoped that in the near future they will be resolved, and their practical application will help humanity enter a new round of technological development.

Hi all!

There is an opinion that it is not profitable to engage in science today - one cannot become rich! But I hope that today's post will show you that this is far from the case. Today I will tell you how, doing basic research, you can earn a tidy sum.

At any stage of development, any of the sciences has always faced a number of unresolved problems and tasks that haunted scientists. Physics is cold thermonuclear fusion, mathematics is the Goldbach hypothesis, medicine is a cure for cancer, and so on. Some of them are so important (for one reason or another) that a reward is due for their solution. And sometimes this reward is very, very decent.

In a number of sciences, the Nobel Prize can serve as this reward. But they don’t give it for mathematical discoveries, and today I would like to talk about mathematics.

Mathematics - the queen of sciences, offers you a sea of ​​unsolved problems and interesting tasks, but today we will talk about only seven. They are also called the Millennium Targets.

It would seem, tasks, and tasks? What is special about them? The fact is that their solution has not been found for many years, and for the solution of each of them, the Clay Institute promised a reward of $ 1 million! Agree, not a lot. Of course, not the Nobel Prize, the size of which is approximately 1.5 million, but it will also do.

Here is their list:

• Equality of classes P and NP
• Hodge hypothesis
• Poincare conjecture (solved)
• Riemann hypothesis
• Quantum Yang-Mills theory
• Existence and smoothness of solutions of the Navier-Stokes equations
• Birch-Swinnerton-Dyer hypothesis

So let's take a closer look at each of them.

1.Equality of classes P and NP

This problem is one of the most important problems in the theory of algorithms, and, I bet, many of you, at least indirectly, have heard about it. What is this problem and what is its essence? Imagine that there is a certain class of problems to which we can quickly give an answer, that is, quickly find a solution for them. This class of problems in the theory of algorithms is called the P class. And there is a class of problems for which we can quickly check the correctness of their solution - this is the NP class. And so far, it is not known whether these classes are equal or not. That is, it is not known whether it is possible, at least in theory, to find such an algorithm by which we can find the solution to the problem as quickly as we can check its correctness.

Classic example. Let a set of numbers be given, for example: 50, 2, 47, 5, 21, 4, 78, 1. Problem: is it possible to choose among these numbers such that their sum will give 100? Answer: you can, for example, 50 + 47 + 2 + 1 = 100. It is easy to check the correctness of the solution. We apply the addition operation four times and that's it. It's just a matter of picking up those numbers. At first glance, this is much more difficult to do. That is, finding a solution to a problem is more difficult than checking it. From the point of view of banal erudition, this is true, but mathematically this has not been proven, and there is hope that this is not so.

And so what? What if it turns out that the classes P and NP are equal? Everything is simple. Class equality means that there are algorithms for solving many problems that work much faster than currently known (as mentioned above).

Naturally, far from one attempt was made to prove or disprove this hypothesis, but none was successful. The latest attempt was made by the Indian mathematician Vinay Deolalikar. According to the author of the problem statement, Stephen Cook, this solution was "a relatively serious attempt to solve the problem of P vs NP". But, unfortunately, a number of errors were found in the presented proof, which the author promised to correct.

2. Hodge hypothesis

The complex is the sum of the simple parts. As a result of studying complex objects, mathematicians have developed methods for their approximation by gluing objects of increasing dimension. But it has not yet been clarified to what extent this kind of approximation can be carried out, and the geometric nature of some objects that are used in the approximation remains unclear.

3.Poincaré's hypothesis

The Poincaré hypothesis is currently the only one of the seven Millennium Challenges that has been solved. It is gratifying to note that our compatriot Grigory Yakovlevich Perelman, part-time reclusive genius, became the author of the decision. You can talk about it a lot and interestingly, but let's focus on the hypothesis itself.

Formulation:

Every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.

Or the generalized Poincare conjecture:

For any natural number n, any manifold of dimension n is homotopy equivalent to a sphere of dimension n if and only if it is homeomorphic to it.

In a simple way, the essence of the problem is as follows. If we take an apple and cover it with a rubber film, then with the help of deformations, without tearing the film, we can turn the apple into a dot or a cube, but in no way can we turn it into a donut. A cube, a 3D sphere, and even 3D space are identical to each other, up to deformation.

Despite such a simple formulation, the hypothesis remained unproven for hundreds of years. Although in mathematics, sometimes, the simpler the formulation, the more difficult the proof (we all remember Fermat's Last Theorem).

Let's return to Comrade Perelman. This gentleman is also famous for the fact that he refused the million put to him, stating the following: “Why do I need your money, if I have the whole Universe in my hands?” I wouldn't be able to do that. As a result of the refusal, the allocated million was granted to young French and American mathematicians.

Finally, I would like to note that the Poincaré hypothesis has absolutely no practical application (!!!).

4. The Riemann hypothesis.

The Riemann Hypothesis is probably the most famous (along with the Poincaré Hypothesis) of the seven Millennium Problems. One of the reasons for its popularity among non-professional mathematicians is that it has a very simple formulation.

All non-trivial zeros of the Riemann zeta function have a real part equal to ?.

Agree, it's quite simple. And the apparent simplicity was the reason for many attempts to prove this hypothesis. Unfortunately, so far to no avail.

A large number of unsuccessful attempts to prove the Riemann hypothesis gave rise to doubts about its validity among some mathematicians. Among them is John Littlewood. But the ranks of skeptics are not so numerous and most of the mathematical community tends to believe that the Riemann hypothesis is, nevertheless, correct. Indirect confirmation of this is the validity of a number of similar statements and hypotheses.

Many algorithms and statements in number theory have been formulated with the assumption that the above conjecture is true. Thus, the proof of the validity of the Riemann hypothesis will confirm the foundation of number theory, and its refutation of number theory will “shake” the very foundation.

And, finally, one rather well-known, but very interesting fact. Once David Gilbert was asked: “What will be your first actions if you sleep for 500 years and wake up?” "I'll ask if the Riemann Hypothesis has been proven."

5. Yang-Mills theory

One of the gauge theories of quantum physics with a non-Abelian gauge group. This theory was proposed in the middle of the last century, but for a long time it was considered as a purely mathematical technique that has nothing to do with the real nature of things. But later, on the basis of the Yang-Mills theory, the main theories of the Standard Model were built - quantum chromodynamics and the theory of weak interactions.

Problem formulation:

For any simple compact gauge group, the quantum Yang-Mills theory for space exists and has a nonzero mass defect.

The theory is perfectly confirmed by the results of experiments and the results of computer simulations, but has not received theoretical proof.

6. Existence and smoothness of solutions of the Navier-Stokes equations

One of the most important problems in hydrodynamics, and the last of the unsolved problems of classical mechanics.

The Navier-Stokes equation, supplemented by Maxwell's equations, heat transfer equations, etc., is used in solving many problems of electrohydrodynamics, magnetohydrodynamics, fluid and gas convection, thermal diffusion, etc.

The equations themselves are a system of partial differential equations. The equations have two parts:

• equations of motion
• continuity equations

Finding a complete analytical solution of the Navier-Stokes equations is greatly complicated by their nonlinearity and strong dependence on boundary and initial conditions.

7. Birch-Swinnerton-Dyer hypothesis

The last of the millennium problems is the Birch-Swinnerton-Dyer hypothesis.

The hypothesis states that

the rank of an elliptic curve r over Q is equal to the order of zero of the Hasse-Weil zeta function

E(L,s) at the point s = 1.

This conjecture is the only relatively simple way to determine the rank of elliptic curves, which, in turn, are the main objects of study in modern number theory and cryptography.

That's all the problems of the millennium. I apologize for the fact that some issues are covered much less than others. This is due to the lack of information on these problems and the impossibility of quite simply (without involving cumbersome and complex mathematics) to state their essence. The Clay Institute offered a $1 million reward for solving each of the problems. Dare! There is a chance to earn good money by moving forward fundamental science, because six out of seven problems have not yet been solved.