Math problems. Unproven theorems of our time, for which a reward is due. Yang-Mills theory

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 now, mathematicians all over the world have been racking their brains to solve them. Those who succeed will receive a reward of one million US dollars, offered by the Clay Institute.

Clay Institute

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

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

Millennium Challenges

The Clay Institute list originally included:

• Hodge cycle hypothesis;
• equations quantum theory Young-Mills;
• Poincaré conjecture;
• problem of equality of classes P and NP;
• Riemann hypothesis;
• about 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 Grigory Perelman proved

In 1900, the famous scientist-philosopher Henri Poincaré proposed that every simply connected compact 3-dimensional manifold without boundary is homeomorphic to a 3-dimensional 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 solving the Poincaré problem. They produced the effect of a bomb exploding. In 2010, the Poincaré hypothesis was excluded from the list of “Unsolved Problems” of the Clay Institute, and Perelman himself was offered to receive the 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 was able to prove can be given by imagining that they stretch a rubber disk over a donut (torus), and then try to pull the edges of its circle to one point. Obviously this is impossible. It's a different matter if you perform this experiment with a ball. In this case, it seems that a 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 ordinary person, but two-dimensional from a mathematical point of view.

Poincaré suggested that the three-dimensional sphere is the only three-dimensional “object” whose surface can be contracted to one 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 zero mass defect.

Speaking in a language that the average person can understand, 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 general theory fields. It must become a tool to explain all these interactions. Yang-Mills theory is mathematical language, with the help of which it became possible to describe 3 of the 4 main forces of nature. It doesn't apply to gravity. Therefore, it cannot be considered 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 under strong coupling.

Navier-Stokes equations

These expressions describe processes such as air currents, fluid flow, and turbulence. For some special cases, analytical solutions to the Navier-Stokes equation have already been found, but no one has yet succeeded in doing this for the general case. At the same time, numerical modeling for specific values ​​of speed, density, pressure, time, and so on allows one to achieve excellent results. We can only hope that someone will be able to apply the Navier-Stokes equations in reverse direction, i.e., calculate the parameters using them, or prove that there is no solution method.

Birch-Swinnerton-Dyer problem

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

If for each prime number we count the number of points on the curve modulo it, we 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 Birch and Peter Swinnerton-Dyer proposed a conjecture regarding elliptic curves. According to it, the structure and quantity of the set of its rational solutions are related to the behavior of the L-function in the unit. Unproven at this moment The Birch-Swinnerton-Dyer conjecture depends on the description of algebraic equations of degree 3 and is the only relatively simple general way to calculate the rank of elliptic curves.

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

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-Lewin problem, can be formulated in clear language as follows. Let's assume that a positive answer to a certain question can be checked quickly enough, that is, in polynomial time (PT). Then is it correct to say that the answer to it can be found fairly quickly? It sounds even simpler: is it really no more difficult to check the solution to a problem than to find it? If the equality of the classes p and np is ever proven, then all selection problems can be solved by 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 dealing with other issues. However, by the middle of the 19th century, the situation changed, and they became one of the most relevant ones that mathematics began to study.

The Riemann hypothesis, which emerged 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, many of the fundamental principles of modern cryptography, which form the basis of much of the electronic commerce mechanisms, will have to be reconsidered.

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, the strength of modern cryptokeys will be questioned.

Hodge cycle conjecture

This still 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 dimension. This method has been known and successfully used for quite a long time. However, it is not known to what extent simplification can be carried out.

Now you know what unsolvable problems exist at the moment. They are the subject of research by thousands of scientists around the world. We can only hope that they will be resolved in the near future, and their practical use will help humanity enter a new stage of technological development.

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 anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.

There are not many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only math problem, which became so widely known and became a real legend. It is mentioned in many books and films, and the main context of almost all mentions is the impossibility of proving the theorem.

Yes, this theorem is very well known 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 essence and understandable to anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.

Why is she so famous? Now we'll find out...

Are there many proven, unproven and as yet unproven theorems? The point here is that Fermat's Last Theorem represents the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult problem, and yet its formulation can be understood by anyone with the 5th grade of high school, but not even every professional mathematician can understand the proof. Neither in physics, nor in chemistry, nor in biology, nor in mathematics, is there a single problem that could be formulated so simply, but remained unsolved 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 triplets satisfying the equality 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 C's and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the brotherhood were more philosophers and aesthetes than mathematicians.

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

Starting from 3, 4, 5 - indeed, a junior 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 trick begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, its absence. When you need to prove that there is a solution, you can and should simply present this solution.

Proving 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! (give solution). And that’s it, the opponent is defeated. How to prove absence?

Say: “I haven’t found such solutions”? Or maybe you weren't looking well? What if they exist, only very large, very large, such that even a super-powerful computer still doesn’t have enough strength? This is what is difficult.

This can be shown visually like this: if you take two squares of suitable sizes and disassemble them into unit squares, then from this bunch of unit squares you get a third square (Fig. 2):


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

But the 17th century mathematician Frenchman Pierre de Fermat enthusiastically studied the general equation x n + y n = z n. And finally, I concluded: for n>2 there are no integer solutions. Fermat's proof is irretrievably lost. 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 Fermat has a reputation for never making mistakes. Even if he did not leave evidence of a statement, it was subsequently confirmed. Moreover, Fermat proved his thesis for n=4. Thus, 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 a proof (in 1770 he proposed a solution for n = 3),

Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the 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 epic of searching for a proof of Fermat’s last theorem was practically over.

It is easily shown that it is enough to prove Fermat’s theorem only for simple 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, female 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 proven for almost all n less than one hundred.

Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that the theorem in general cannot be proven using the methods of mathematics of the 19th century. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unawarded.

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

He soon died of natural causes. The heirs were quite 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 Wolfskehl Prize. 100,000 marks were awarded to the person who proved Fermat's theorem. Not a pfennig was awarded for refuting the theorem...

Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and resolutely refused to waste time on such a useless exercise. But the amateurs had a blast. A few weeks after the announcement, an avalanche of “evidence” hit the University of Göttingen. Professor E.M. Landau, whose responsibility was to analyze the evidence sent, distributed cards to his students:

Dear. . . . . . . .

Thank you for sending me the manuscript with the proof of Fermat’s Last Theorem. The first error is on page ... in line... . Because of it, the entire proof loses its validity.
Professor E. M. Landau

In 1963, Paul Cohen, relying 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 undecidable?! But true Great Theorem fanatics were not disappointed at all. The advent of computers suddenly gave mathematicians a new method of proof. After World War II, teams 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 1980s, Samuel Wagstaff raised the limit to 25,000, and in the 1990s, mathematicians declared that Fermat's Last Theorem was true for all values ​​of n up 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 began researching modular forms. These forms generate series of numbers, each with its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. No connection has ever been found between such different objects.

However, 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 an entire direction in mathematics, but until the Taniyama-Shimura hypothesis was proven, 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 could not have a counterpart in the modular world. From now on, Fermat's Last Theorem was inextricably linked with the Taniyama-Shimura conjecture. Having proven 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 proven. But for thirty years it was not possible to prove the Taniyama-Shimura hypothesis, and there was less and less hope for success.

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

Having learned about Ken Ribet's findings, Wiles plunged headlong into proving the Taniyama-Shimura hypothesis. He decided to work in complete isolation and secrecy. “I realized that everything that had anything to do with Fermat’s Last Theorem arouses too much interest... Too many spectators obviously 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 paper 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. Every piece of evidence must be carefully examined before the evidence can be considered rigorous and accurate. Wiles spent a restless summer waiting for feedback from reviewers, hoping that he would be able to win their approval. At the end of August, experts found the judgment to be insufficiently substantiated.

It turned out that this decision contained a gross error, although in general it was correct. Wiles did not give up, called on the help of the famous specialist in number theory Richard Taylor, and already in 1994 they published a corrected and expanded proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the mathematical journal “Annals of Mathematics”. But the story did not end there either - the final point was reached only 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 festive dinner on the occasion of her birthday, I presented Nadya with the manuscript of the complete proof” (Andrew Wales). Have I not yet said that mathematicians are strange people?

This time there was no doubt about the evidence. 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 that Fermat’s Last Theorem is unsolvable. But even those who know about the proof found continue to work in this direction - few are satisfied that the Great Theorem requires a solution of 130 pages!

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

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1. 1 Murad:

   We considered the equality Zn = Xn + Yn to be Diophantus’s equation or Fermat’s great theorem, and this is the solution to 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 whole numbers+X – even, last right digits: 0, 2, 4, 6, 8 and +Y – odd, last right digits: 1, 3, 5, 7, 9, i.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 is +Z = +X + Y= +(X + Y).
   Integers -Z consist of the union of -X – even and -Y – odd, and satisfy the equation:
   (Z + X) X = (Z + Y) Y, and the solution is -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 are made up 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 also the solutions R2 = X2 + Y2 and R2 =- (X2 + Y2) are circles, the centers are the origin of the square coordinate system and with radius R. They can be written in the form (5n)2 = (3n)2 + (4n)2 , where n are positive and negative integers, and are 3 consecutive numbers. Also the solutions are 2 -bit numbers XY, which starts with 00 and ends with 99 and is 102 = 10x10 and count 1 century = 100 years.
   Let's consider solutions when n = 3. Then Z3 = X3 + Y3 solutions to the equation (Z3 – X3) X3 = (Z3 – Y3) Y3.
   3-digit numbers XYZ starts with 000 and ends with 999 and is 103 = 10x10x10 = 1000 years = 10 centuries
   From 1000 cubes of the same size and color, you can make a rubik of the order of 10. Consider a rubik of the order +103=+1000 - red and -103=-1000 - blue. They consist of 103 = 1000 cubes. If we lay it out and put the cubes in one row or on top of each other, without gaps, we will get a horizontal or vertical segment of length 2000. Rubik is a large cube, covered with small cubes, starting from size 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 is 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 on 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) 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 holds: 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 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, 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 in given number add and subtract the number b, then the sum does not change, but the product does, 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 we put integers instead of the letters a and b, we get paradoxes, absurdities, and mistrust of mathematics.