Sometimes diligent study of the exact sciences can bear fruit - you will become not only known to the whole world, but also rich. Awards are given, however, not for anything, and in modern science there are a lot of unproven theories, theorems and problems that multiply as the sciences develop, take, for example, the Kourovka or Dniester notebooks, a sort of collection of unsolvable physics and mathematics, and not only, tasks. However, there are also truly complex theorems that have not been able to solve for more than a dozen years, and it is for them that the American Clay Institute has been awarded an award in the amount of 1 million US dollars each. Until 2002, the total jackpot was 7 million, since there were seven Millennium Problems, but Russian mathematician Grigory Perelman solved Poincaré's hypothesis by epically abandoning a million, without even opening the door to US mathematicians who wanted to give him his honestly earned bonus. So, we turn on the Big Bang Theory for the background and mood, and see what else you can cut a round sum for.

Equality of classes P and NP

In simple terms, the P = NP equality problem is as follows: if a positive answer to some question can be checked fairly quickly (in polynomial time), then is it true that the answer to this question can be found rather quickly (also in polynomial time and using polynomial memory)? In other words, is it really no easier to check the solution to the problem than to find it? The bottom line is that some calculations and calculations are easier to solve by an algorithm, rather than brute force, and thus save a lot of time and resources.

Hodge hypothesis

The Hodge conjecture was formulated in 1941 and is that for especially good types of spaces, called projective algebraic varieties, the so-called Hodge cycles are combinations of objects that have a geometric interpretation - algebraic cycles.

Here, explaining in simple words, we can say the following: in the 20th century, very complex geometric shapes, such as curved bottles, were discovered. So, it was suggested that in order to construct these objects for description, it is necessary to use completely puzzling forms that do not have the geometric essence of "such scary multidimensional malyaks" or you can still get by with conventionally standard algebra + geometry.

Riemann hypothesis

Here it is rather difficult to explain in human language, it is enough to know that the solution of this problem will have far-reaching consequences in the field of the distribution of primes. The problem is so important and urgent that even the derivation of a counterexample of the hypothesis is at the discretion of the academic council of the university, the problem can be considered proven, so here you can try the method "from the opposite". Even if it is possible to reformulate the hypothesis in a narrower sense, then the Clay Institute will pay a certain amount of money.

Young - Mills theory

Particle physics is one of the favorite areas of Dr. Sheldon Cooper. Here the quantum theory of two smart guys tells us that for any simple gauge group in space there is a mass defect other than zero. This statement was established by experimental data and numerical modeling, but no one can yet prove it.

Navier-Stokes equations

Here Howard Wolowitz would probably help us if he existed in reality - after all, this is a riddle from hydrodynamics, and the basis of the foundations. The equations describe the motion of a viscous Newtonian fluid, are of great practical importance, and most importantly describe turbulence, which cannot be driven into the framework of science and predicted its properties and actions. Justification of the construction of these equations would allow not to poke a finger in the sky, but to understand turbulence from the inside and make aircraft and mechanisms more stable.

Birch - Swinnerton-Dyer hypothesis

Here I, however, tried to find simple words, but there is such dense algebra that one cannot do without deep immersion. For those who do not want to scuba dive in matan, you need to know that this hypothesis allows you to quickly and painlessly find the rank of elliptic curves, and if this hypothesis did not exist, then a sheet of calculations would be needed to calculate this rank. Well, of course, you also need to know that the proof of this hypothesis will enrich you by a million dollars.

It should be noted that there are already advances in almost every area, and even cases have been proven for individual examples. Therefore, do not hesitate, otherwise it will turn out as with Fermat's theorem, which succumbed to Andrew Wiles after more than 3 centuries in 1994, and brought him the Abel Prize and about 6 million Norwegian crowns (50 million rubles at today's exchange rate).

1. 1 Murad:

   We the equality Zn = Xn + Yn considered the Diophantus equation or the Great Fermat's theorem, and this is the solution of the equation (Zn-Xn) Xn = (Zn - Yn) Yn. Then Zn = - (Xn + Yn) is a solution to the equation (Zn + Xn) Xn = (Zn + Yn) Yn. These equations and solutions are related to the properties of integers and actions on them. So we don't know the properties of integers ?! With such limited knowledge, we will not reveal the truth.
   Consider the solutions Zn = + (Xn + Yn) and Zn = - (Xn + Yn) when n = 1. Integers + Z are formed using 10 digits: 0, 1, 2, 3, 4, 5, 6, 7 , 8, 9. They are divided by 2 integers + X - even, the last right digits: 0, 2, 4, 6, 8 and + Y - odd, the last right digits: 1, 3, 5, 7, 9, t .e. + X = + Y. The number of Y = 5 - odd and X = 5 - even numbers is: Z = 10. Satisfies the equation: (Z - X) X = (Z - Y) Y, and the solution + Z = + X + Y = + (X + Y).
   The integers -Z are composed of the concatenation of -X - even and -Y - odd, and satisfy the equation:
   (Z + X) X = (Z + Y) Y and solution -Z = - X - Y = - (X + Y).
   If Z / X = Y or Z / Y = X, then Z = XY; Z / -X = -Y or Z / -Y = -X, then Z = (-X) (- Y). Division is verified by multiplication.
   One-digit positive and negative numbers are composed of 5 odd and 5 odd numbers.
   Consider the case n = 2. Then Z2 = X2 + Y2 is a solution to the equation (Z2 - X2) X2 = (Z2 - Y2) Y2 and Z2 = - (X2 + Y2) is a solution to the equation (Z2 + X2) X2 = (Z2 + Y2) Y2. We considered Z2 = X2 + Y2 to be the Pythagorean theorem, and then the solution Z2 = - (X2 + Y2) is the same theorem. We know that the diagonal of a square is divided into 2 parts, where the diagonal is the hypotenuse. Then the equalities are true: Z2 = X2 + Y2, and Z2 = - (X2 + Y2) where X and Y are legs. And also solutions R2 = X2 + Y2 and R2 = - (X2 + Y2) are circles, centers are the origin of a square coordinate system and with a radius R. They can be written as (5n) 2 = (3n) 2 + (4n) 2 where n are positive and negative integers and are 3 consecutive numbers. Also solutions are 2-bit XY numbers that start at 00 and end at 99 and are 102 = 10x10 and count 1 century = 100 years.
   Consider solutions when n = 3. Then Z3 = X3 + Y3 are solutions to the equation (Z3 - X3) X3 = (Z3 - Y3) Y3.
   3-digit numbers XYZ starts at 000 and ends at 999 and is 103 = 10x10x10 = 1000 years = 10 centuries
   From 1000 cubes of the same size and color, you can make a rubik of about 10. Consider a rubik of about + 103 = + 1000 - red and -103 = -1000 - blue. They consist of 103 = 1000 cubes. If we expand, and put the cubes in one row or on top of each other, without gaps, then we get a horizontal or vertical segment of length 2000. Rubik is a large cube, covered with small cubes, starting with a size of 1butto = 10st.-21, and cannot be added to it or subtract one cube.
   - (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9+10); + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9+10);
   - (12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92+102); + (12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92+102);
   - (13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93+103); + (13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93+103).
   Each integer 1. Add 1 (units) 9 + 9 = 18, 10 + 9 = 19, 10 +10 = 20, 11 +10 = 21, and the products:
   111111111 x 111111111 = 12345678987654321; 1111111111 x 111111111 = 123456789987654321.
   0111111111x1111111110 = 0123456789876543210; 01111111111x1111111110 = 01234567899876543210.
   These operations can be performed with 20-bit calculators.
   It is known that + (n3 - n) is always divisible by +6, and - (n3 - n) is always divisible by -6. We know that n3 - n = (n-1) n (n + 1). These are 3 consecutive numbers (n-1) n (n + 1), where n is even, then divisible by 2, (n-1) and (n + 1) are odd, divisible by 3. Then (n-1) n (n + 1) is always divisible by 6. If n = 0, then (n-1) n (n + 1) = (- 1) 0 (+1), n ​​= 20, then (n-1) n (n + 1) = (19) (20) (21).
   We know that 19 x 19 = 361. This means that one square is surrounded by 360 squares and then one cube is surrounded by 360 cubes. The equality is fulfilled: 6 n - 1 + 6n. If n = 60, then 360 - 1 + 360, and n = 61, then 366 - 1 + 366.
   Generalizations follow from the above statements:
   n5 - 4n = (n2-4) n (n2 + 4); n7 - 9n = (n3-9) n (n3 + 9); n9 -16 n = (n4-16) n (n4 + 16);
   0 ... (n-9) (n-8) (n-7) (n-6) (n-5) (n-4) (n-3) (n-2) (n-1) n (n +1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6) (n + 7) (n + 8) (n + 9)… 2n
   (n + 1) x (n + 1) = 0123 ... (n-3) (n-2) (n-1) n (n + 1) n (n-1) (n-2) (n-3 ) ... 3210
   n! = 0123 ... (n-3) (n-2) (n-1) n; n! = n (n-1) (n-2) (n-3) ... 3210; (n + 1)! = n! (n +1).
   0 +1 + 2 + 3 + ... + (n-3) + (n-2) + (n-1) + n = n (n + 1) / 2; n + (n-1) + (n-2) + (n-3) + ... + 3 + 2 + 1 + 0 = n (n + 1) / 2;
   n (n + 1) / 2 + (n + 1) + n (n + 1) / 2 = n (n + 1) + (n + 1) = (n + 1) (n + 1) = (n +1) 2.
   If 0123 ... (n-3) (n-2) (n-1) n (n + 1) n (n-1) (n-2) (n-3)… 3210 х 11 =
   = 013… (2n-5) (2n-3) (2n-1) (2n + 1) (2n + 1) (2n-1) (2n-3) (2n-5)… 310.
   Any integer n is powers of 10, has: - n and + n, + 1 / n and -1 / n, odd and even:
   - (n + n + ... + n) = -n2; - (n x n x ... x n) = -nn; - (1 / n + 1 / n + ... + 1 / n) = - 1; - (1 / n x 1 / n x ... x1 / n) = -n-n;
   + (n + n + ... + n) = + n2; + (n x n x ... x n) = + nn; + (1 / n + ... + 1 / n) = + 1; + (1 / n x 1 / n x… x1 / n) = + n-n.
   It is clear that if any integer number is added by itself, then it will increase 2 times, and the product will be a square: X = a, Y = a, X + Y = a + a = 2a; XY = a x a = a2. This was considered Vieta's theorem - a mistake!
   If you add and subtract the number b to this number, then the sum does not change, but the product changes, for example:
   X = a + b, Y = a - b, X + Y = a + b + a - b = 2a; XY = (a + b) x (a –b) = a2- b2.
   X = a + √b, Y = a -√b, X + Y = a + √b + a - √b = 2a; XY = (a + √b) x (a -√b) = a2- b.
   X = a + bi, Y = a - bi, X + Y = a + bi + a - bi = 2a; XY = (a + bi) x (a –bi) = a2 + b2.
   X = a + √b i, Y = a - √bi, X + Y = a + √bi + a - √bi = 2a, XY = (a -√bi) x (a -√bi) = a2 + b.
   If instead of the letters a and b we put integers, then we get paradoxes, absurdities, and mistrust of mathematics.

There are not so many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only mathematical problem that has received such wide popularity and has become a real legend. It is mentioned in many books and films, while the main context of almost all references is the impossibility of proving the theorem.

Yes, this theorem is very famous and, in a sense, has become an "idol" worshiped by amateur mathematicians and professionals, but few people know that its proof has been found, and it happened back in 1995. But first things first.

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 essence and understandable to any person with a secondary education. It says that the formula a to the power n + b to the power n = c to the power n has no natural (that is, non-fractional) solutions for n> 2. It seems that everything is simple and clear, but the best mathematicians and ordinary amateurs fought over seeking a solution for more than three and a half centuries.

Why is she so famous? We'll find out now ...

Are there few proven, unproven, and not yet proven theorems? The point is that Fermat's Last Theorem is the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and nevertheless its formulation can be understood by everyone with the 5th grade of high school, but the proof is not even every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics, there is not a single problem that would be formulated so simply, but remained unsolved for so long. 2. What does it consist of?

Let's start with the 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-angled 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 triples of integers satisfying the equality x² + y² = z². They proved that there are infinitely many Pythagorean triplets, and received 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 useless attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.

That is, it is easy to find 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.

So, it turns out that they are NOT. This is where the catch begins. 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, you can and should just give this solution.

Proving the absence is more difficult: 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! (please provide a solution). And that's it, the opponent is slain. How to prove absence?

Say, "I have not found such solutions"? Or maybe you were looking badly? What if they are, only very large, well, very, such that even a super-powerful computer does not have enough strength yet? This is what is difficult.

In a visual form, this can be shown as follows: if you take two squares of suitable sizes and disassemble into unit squares, then from this heap of unit squares you get the third square (Fig. 2):


And if we do the same with the third dimension (Fig. 3), it will not work. Not enough cubes, or extra ones remain:

But the mathematician of the 17th century, Frenchman Pierre de Fermat, enthusiastically studied the general equation x n + y n = z n. And finally, I came to the conclusion: there are no integer solutions for n> 2. Fermat's proof is irretrievably lost. The manuscripts are burning! 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 contain it."

Actually, a theorem without proof is called a hypothesis. But for Fermat the fame was fixed that he was never wrong. Even if he did not leave evidence 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, such great minds as Leonard Euler worked on the search for 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 mathematicians saw and believed that the three-century saga of searching for a proof of Fermat's last theorem was practically over.

It is easy to show that it is sufficient 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 also infinitely many primes ...

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

Finally, the German mathematician Ernst Kummer showed in a brilliant study that it is impossible to prove the theorem in general form using the methods of 19th century mathematics. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, was not awarded.

In 1907, the wealthy German industrialist Paul Wolfskel, out of unrequited love, decided to commit suicide. As a true German, he set the date and time for the suicide: exactly at midnight. On the last day, he drew up a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Out of nothing to do, he went to the library and began to read the famous article by Kummer. Suddenly it seemed to him that Kummer made a mistake in the course of his reasoning. Wolfskel began to sort through this passage of the article, pencil in hand. Midnight has passed, morning has come. The gap in the evidence 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 a natural death. The heirs were quite surprised: 100,000 marks (more than 1,000,000 of the 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 Wolfskehl Prize. 100,000 marks were due to the prover of Fermat's theorem. Not a pfennig was supposed to refute the theorem ...

Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and strongly refused to waste time on such a useless exercise. But the amateurs frolicked wonderfully. 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 submitted evidence, handed out cards to his students:

Dear. ... ... ... ... ... ... ...

Thank you for the manuscript you sent me with the proof of Fermat's Last Theorem. The first error is on page ... in line .... Because of it, all evidence is null and void.
Professor E. M. Landau

In 1963, Paul Cohen, relying on Gödel's conclusions, proved the undecidability of one of Hilbert's twenty-three problems - the continuum hypothesis. What if Fermat's Last Theorem is also undecidable ?! But the true fanatics of the Great Theorem were not disappointed in the least. 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 declared that Fermat's Last Theorem was true for all values ​​of n to 4 million. But if you subtract even a trillion trillion from infinity, it will not become smaller. Mathematicians are not convinced by statistics. To prove the Great Theorem meant to prove it for ALL n going to infinity.

In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate rows of numbers, each with its own row. By chance Taniyama compared these series with the series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. Connections have never been found between such different objects.

Nevertheless, friends, after careful testing, put forward a hypothesis: each elliptic equation has a double - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole direction in mathematics, but until the Taniyama-Shimura hypothesis was proved, the entire 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 an elliptic equation with a solution to Fermat's equation does not exist, and Fermat's Last Theorem would be proved immediately. But for thirty years, it was not possible to prove the Taniyama-Shimura hypothesis, and there were fewer and fewer 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 went headlong into proving the Taniyama-Shimura hypothesis. 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 bore fruit, 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.), On which work lasted more than seven years.

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

It turned out that this solution contains a gross error, although on the whole it is correct. Wiles did not give up, called for the help of a well-known expert 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 as many as 130 (!) Pages in the mathematical journal "Annals of Mathematics". But the story did not end there either - the last point was only put in the next 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 gala dinner on the occasion of her birthday, I presented Nadia with the manuscript of the complete proof” (Andrew Waltz). Have I said that mathematicians are strange people?

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

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

Therefore, now the forces of very many mathematicians (mostly amateurs, not professional scientists) are thrown into the search for a simple and laconic proof, but this path, most likely, will not lead anywhere ...

a source

Unsolvable problems are 7 interesting mathematical problems. Each of them was proposed at one time by famous scientists, usually in the form of hypotheses. For many decades, mathematicians all over the world have been puzzling over their solution. Those who succeed will be rewarded with a million American dollars, offered by the Clay Institute.

Clay Institute

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

In the early 21st century, the Clay Mathematical Institute offered an award to those who solve what are known as the most difficult unsolvable problems, calling their list the Millennium Prize Problems. From the "Hilbert's List" only the Riemann hypothesis was included in it.

Millennium Challenges

The Clay Institute's list originally included:

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

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

What Grigory Perelman proved

In 1900, the famous scientist-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 has not been found for a century. Only in 2002-2003 the St. Petersburg mathematician G. Perelman published a number of articles on the solution of the Poincaré problem. They had the effect of a bomb exploding. In 2010, Poincaré's hypothesis was excluded from the list of "Unsolved Problems" of the Clay Institute, and Perelman himself was asked to receive a considerable reward 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 over a donut (torus), and then they are trying to pull the edges of its circle into one point. This is obviously not possible. It's another matter if you perform this experiment with a ball. In this case, a seemingly three-dimensional sphere, resulting from a disk, the circumference of which has been pulled into a point by a hypothetical cord, will be three-dimensional in the understanding of an ordinary person, but two-dimensional in terms of mathematics.

Poincaré suggested that a three-dimensional sphere is the only three-dimensional "object", the surface of which can be contracted to one point, and Perelman was able to prove this. Thus, the list of "Unsolvable tasks" 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 space theory created by Yang and Mills exists and has zero mass defect.

If we speak in a language understandable for an ordinary person, 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 the help of which it became possible to describe 3 of the 4 basic forces of nature. It does not apply to gravity. Therefore, it cannot be assumed that Young 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 perturbation theory series. 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 currents, fluid flow, and turbulence. For some special cases, analytical solutions of the Navier-Stokes equation have already been found, but 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 hope that someone will be able to apply the Navier-Stokes equations in the opposite direction, that is, to calculate the parameters with their help, or to prove that there is no solution method.

Birch - Swinnerton-Dyer problem

The category of "Unsolved problems" includes the hypothesis proposed by British scientists from the University of Cambridge. As early as 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, you count the number of points on the curve modulo its modulus, you get an infinite set of integers. If you specifically "glue" it into 1 function of a complex variable, then you get the Hasse-Weil zeta function for a curve of the third order, denoted by the letter L. It contains information about the behavior modulo all primes at once.

Brian Birch and Peter Swinnerton-Dyer hypothesized about elliptic curves. According to her, the structure and number of the set of its rational decisions are associated with the behavior of the L-function at unity. The currently unproven Birch - Swinnerton-Dyer conjecture depends on the description of algebraic equations of degree 3 and is the only relatively simple general method for calculating the rank of elliptic curves.

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

Equality of classes p and np

If the rest of the Millennium Problems are purely mathematical, then this one is related to the current theory of algorithms. The problem concerning the equality of the classes p and np, also known as the Cook-Levin problem, can be easily formulated as follows. Suppose that an affirmative answer to a certain question can be verified quickly enough, that is, in polynomial time (PV). Then is it correct to say that the answer to it can be found rather quickly? It sounds even simpler: is it really no more difficult to check a solution to a problem than to find it? If the equality of the classes p and np is ever proved, then all selection problems can be solved in a 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 was engaged in other issues. However, by the middle of the 19th century, the situation had changed, and they became one of the most relevant in which mathematicians began to study.

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

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

According to the Riemann hypothesis, the nature of the distribution of primes may be significantly different from what is currently assumed. The fact is that until now 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 primes 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 strength of modern crypto keys will be called into question.

Hodge cycles hypothesis

This still unsolved problem was formulated in 1941. The Hodge hypothesis assumes the possibility of approximating the shape of any object by "gluing" together simple bodies of higher dimension. This method was known and successfully applied 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 solved, and their practical application will help humanity to enter a new round of technological development.