1. 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.

TASKS OF MATHEMATICS UNSOLVED BY HUMANITY

Hilbert problems

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

Landau problems

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

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

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

Millennium targets (Millennium Prize Problems

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

Here is a list of these seven tasks:

No. 1. Equality of classes P and NP

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

No. 2. Hodge hypothesis

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

No. 3. The Poincaré hypothesis (proved by G.Ya. Perelman)

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

No. 4. Riemann hypothesis

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

No. 5. Yang-Mills theory

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

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

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

No. 7. Birch-Swinnerton-Dyer hypothesis

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

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.