Sometimes a 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, for nothing, and in modern science there are a lot of unproven theories, theorems and problems that multiply as science develops, take at least Kourovka or Dniester notebooks, sort of collections with unsolvable physical and mathematical, and not only, tasks. However, there are also truly complex theorems that have not been solved for more than a dozen years, and for them the American Clay Institute has put up an award in the amount of 1 million US dollars for each. Until 2002, the total jackpot was 7 million, since there were seven "millennium problems", but the Russian mathematician Grigory Perelman solved the Poincare conjecture by epically abandoning a million, without even opening the door to US mathematicians who wanted to give him his honestly earned bonuses. 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 equality problem P = NP 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 fairly quickly (also in polynomial time and using polynomial memory)? In other words, is it really not easier to check the solution of the problem than to find it? The bottom line here is that some calculations and calculations are easier to solve algorithmically rather than brute-force, and thus save a lot of time and resources.
Hodge hypothesis
Hodge's conjecture, formulated in 1941, is that for particularly 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 terms, we can say the following: in the 20th century, very complex geometric shapes were discovered, such as curved bottles. 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 “such terrible multidimensional scribbles-scribbles” or you can still get by with conditionally standard algebra + geometry.
Riemann hypothesis
It is quite difficult to explain here in human language, it is enough to know that the solution of this problem will have far-reaching consequences in the field of distribution of prime numbers. 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 also try the “from the opposite” method. Even if it is possible to reformulate the hypothesis in a narrower sense, even here the Clay Institute will pay out a certain amount of money.
Yang-Mills theory
Particle Physics is one of Dr. Sheldon Cooper's favorite topics. Here the quantum theory of two smart uncles tells us that for any simple gauge group in space there is a mass defect other than zero. This statement has been established by experimental data and numerical simulations, but so far no one can prove it.
Navier-Stokes equations
Here, Howard Wolowitz would certainly help us if he existed in reality - after all, this is a riddle from hydrodynamics, and the foundation of the foundations. The equations describe the motions of a viscous Newtonian fluid, are of great practical importance, and most importantly describe turbulence, which cannot be driven into the framework of science in any way and its properties and actions cannot be predicted. Justification for the construction of these equations would allow not to point a finger at the sky, but to understand turbulence from the inside and make aircraft and mechanisms more stable.
Birch-Swinnerton-Dyer hypothesis
True, here I tried to pick up simple words, but there is such a dense algebra that one cannot do without deep immersion. Those who do not want to scuba dive into the matan 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 in almost every area there are already advances, and even proven cases for individual examples. Therefore, do not hesitate, otherwise it will turn out like 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 kroner (50 million rubles at today's exchange rate).
Often, when talking with high school students about research work in mathematics, I hear the following: "What new things can be discovered in mathematics?" But really: maybe all the great discoveries have been made, and the theorems have been proven?
On August 8, 1900, at the International Congress of Mathematicians in Paris, mathematician David Hilbert outlined a list of problems that he believed were to be solved in the twentieth century. There were 23 items on the list. Twenty-one of them have been resolved so far. The last solved problem on Gilbert's list was Fermat's famous theorem, which scientists couldn't solve for 358 years. In 1994, the Briton Andrew Wiles proposed his solution. It turned out to be true.Following the example of Gilbert at the end of the last century, many mathematicians tried to formulate similar strategic tasks for the 21st century. One such list was made famous by Boston billionaire Landon T. Clay. In 1998, at his expense, the Clay Mathematics Institute was founded in Cambridge (Massachusetts, USA) and prizes were established for solving a number of important problems in modern mathematics. On May 24, 2000, the institute's experts chose seven problems - according to the number of millions of dollars allocated for prizes. The list is called the Millennium Prize Problems:
1. Cook's problem (formulated in 1971)
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. This suggests that solving a problem often takes more time than checking the correctness of the solution.
Stephen Cook formulated the problem: can checking the correctness of a solution to a problem be longer than getting the solution itself, regardless of the verification algorithm. This problem is also one of the unsolved problems in the field of logic and computer science. Its solution could revolutionize the fundamentals of cryptography used in the transmission and storage of data.
2. The Riemann Hypothesis (formulated in 1859)
Some integers cannot be expressed as the product of two smaller integers, such as 2, 3, 5, 7, and so on. Such numbers are called prime numbers and play an important role in pure mathematics and its applications. The distribution of prime numbers among the series of all natural numbers does not follow any regularity. However, the German mathematician Riemann made an assumption regarding the properties of a sequence of prime numbers. If the Riemann Hypothesis is proven, it will revolutionize our knowledge of encryption and lead to unprecedented breakthroughs in Internet security.
3. Birch and Swinnerton-Dyer hypothesis (formulated in 1960)
Associated with the description of the set of solutions of some algebraic equations in several variables with integer coefficients. An example of such an equation is the expression x2 + y2 = z2. Euclid gave a complete description of the solutions to this equation, but for more complex equations, finding solutions becomes extremely difficult.
4. Hodge hypothesis (formulated in 1941)
In the 20th century, mathematicians discovered a powerful method for studying the shape of complex objects. The main idea is to use simple "bricks" instead of the object itself, which are glued together and form its likeness. The Hodge hypothesis is connected with some assumptions about the properties of such "bricks" and objects.
5. The Navier - Stokes equations (formulated in 1822)
If you sail in a boat on the lake, then waves will arise, and if you fly in an airplane, turbulent currents will arise in the air. It is assumed that these and other phenomena are described by equations known as the Navier-Stokes equations. The solutions of these equations are unknown, and it is not even known how to solve them. It is necessary to show that the solution exists and is a sufficiently smooth function. The solution of this problem will make it possible to significantly change the methods of carrying out hydro- and aerodynamic calculations.
6. Poincare problem (formulated in 1904)
If you stretch a rubber band over an apple, then you can slowly move the tape without leaving the surface, compress it to a point. On the other hand, if the same rubber band is properly stretched around the donut, there is no way to compress the band to a point without tearing the band or breaking the donut. The surface of an apple is said to be simply connected, but the surface of a donut is not. It turned out to be so difficult to prove that only the sphere is simply connected that mathematicians are still looking for the correct answer.
7. Yang-Mills equations (formulated in 1954)
The equations of quantum physics describe the world of elementary particles. Physicists Yang and Mills, having discovered the connection between geometry and elementary particle physics, wrote their own equations. Thus, they found a way to unify the theories of electromagnetic, weak and strong interactions. From the Yang-Mills equations, the existence of particles followed, which were actually observed in laboratories all over the world, therefore the Yang-Mills theory is accepted by most physicists, despite the fact that within this theory it is still not possible to predict the masses of elementary particles.
I think that this material published on the blog is interesting not only for students, but also for schoolchildren who are seriously involved in mathematics. There is something to think about when choosing topics and areas of research.
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-Weil zeta function for a third-order curve, denoted by the letter L. It contains information about the modulo behavior of 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.
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 become so widely known and has become a real legend. It is mentioned in many books and films, while the main context of almost all mentions is the impossibility of proving the theorem.
Yes, this theorem is very famous and in a sense has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this 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 nature 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 ordinary 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 5 grades 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, 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 + y n \u003d z n. 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, such great minds as Leonhard 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 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 form. 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 hypothesis. 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 hypothesis, 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 ...
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1 Murad :
We considered the equality Zn = Xn + Yn to be the Diophantus equation or Fermat's Great 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 operations 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 divisible by 2 integers +X - even, last right digits: 0, 2, 4, 6, 8 and +Y - odd, 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).
Integers -Z consist of the union of -X for even and -Y for odd, and satisfies the equation:
(Z + X) X = (Z + Y) Y, and the 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 checked by multiplication.
Single-digit positive and negative numbers consist 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 divides it into 2 parts, where the diagonal is the hypotenuse. Then the equalities are valid: Z2 = X2 + Y2, and Z2 = -(X2 + Y2) where X and Y are legs. And more solutions R2 = X2 + Y2 and R2 =- (X2 + Y2) are circles, centers are the origin of the square coordinate system and with 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 starts at 00 and ends at 99 and is 102 = 10x10 and count 1 century = 100 years.
Consider solutions when n = 3. Then Z3 = X3 + Y3 are solutions of the equation (Z3 – X3) X3 = (Z3 – Y3) Y3.
3-bit 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 the order +103=+1000 - red and -103=-1000 - blue. They consist of 103 = 1000 cubes. If we decompose and put the cubes in one row or on top of each other, without gaps, we get a horizontal or vertical segment of length 2000. Rubik is a large cube, covered with small cubes, starting from the size 1butto = 10st.-21, and you cannot add 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 is 1. Add 1(ones) 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 on 20-bit calculators.
It is known that +(n3 - n) is always divisible by +6, and - (n3 - n) is divisible by -6. We know that n3 - n = (n-1)n(n+1). This is 3 consecutive numbers (n-1)n(n+1), where n is even, then divisible by 2, (n-1) and (n+1) 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.
The following 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 x 11=
= 013… (2n-5) (2n-3) (2n-1) (2n+1) (2n+1) (2n-1) (2n-3) (2n-5)…310.
Any integer n is a power 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 is added to itself, then it will increase by 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 we add and subtract the number b to the given number, then the sum does not change, but the product changes, for example:
X \u003d a + b, Y \u003d a - b, X + Y \u003d a + b + a - b \u003d 2a; XY \u003d (a + b) x (a -b) \u003d a2-b2.
X = a +√b, Y = a -√b, X+Y = a +√b + a – √b = 2a; XY \u003d (a + √b) x (a - √b) \u003d a2- b.
X = a + bi, Y = a - bi, X + Y = a + bi + a - bi = 2a; XY \u003d (a + bi) x (a -bi) \u003d 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 we put integer numbers instead of letters a and b, then we get paradoxes, absurdities, and mistrust of mathematics.
