Pierre Fermat, reading the “Arithmetic” of Diophantus of Alexandria and reflecting on its problems, had the habit of writing down the results of his reflections in the form of brief comments in the margins of the book. Against the eighth problem of Diophantus in the margins of the book, Fermat wrote: " On the contrary, it is impossible to decompose either a cube into two cubes, or a biquadrate into two biquadrates, and, in general, no power greater than a square into two powers with the same exponent. I have discovered a truly wonderful proof of this, but these fields are too narrow for it» / E.T. Bell "The Creators of Mathematics". M., 1979, p.69/. I bring to your attention an elementary proof of Fermat’s theorem, which any high school student who is interested in mathematics can understand.
Let us compare Fermat's commentary on Diophantus's problem with the modern formulation of Fermat's last theorem, which has the form of an equation.
« The equation
x n + y n = z n(where n is an integer greater than two)
has no solutions in positive integers»
The comment is in a logical connection with the task, similar to the logical connection of the predicate with the subject. What is asserted by Diophantus' problem is, on the contrary, asserted by Fermat's commentary.
Fermat's comment can be interpreted as follows: if quadratic equation with three unknowns has an infinite number of solutions on the set of all triplets of Pythagorean numbers, then, conversely, an equation with three unknowns to a power greater than the square
There is not even a hint in the equation of its connection with Diophantus' problem. His statement requires proof, but there is no condition from which it follows that it has no solutions in positive integers.
The options for proving the equation known to me boil down to the following algorithm.
- The equation of Fermat's theorem is taken as its conclusion, the validity of which is verified through proof.
- This same equation is called original equation from which its proof must proceed.
As a result, a tautology was formed: “ If an equation has no solutions in positive integers, then it has no solutions in positive integers"The proof of the tautology is obviously incorrect and devoid of any meaning. But it is proven by contradiction.
- An assumption is made that is the opposite of what is stated by the equation that needs to be proven. It should not contradict the original equation, but it does. It makes no sense to prove what is accepted without proof, and to accept without proof what needs to be proven.
- Based on the accepted assumption, absolutely correct mathematical operations and actions are performed to prove that it contradicts the original equation and is false.
Therefore, for 370 years now, proving the equation of Fermat’s Last Theorem has remained an unrealizable dream for specialists and mathematics enthusiasts.
I took the equation as the conclusion of the theorem, and the eighth problem of Diophantus and its equation as the condition of the theorem.
"If the equation x 2 + y 2 = z 2
(1) has an infinite number of solutions on the set of all triples of Pythagorean numbers, then, conversely, the equation x n + y n = z n
, Where n > 2
(2) has no solutions on the set of positive integers.”
Proof.
A) Everyone knows that equation (1) has an infinite number of solutions on the set of all triples of Pythagorean numbers. Let us prove that not a single triple of Pythagorean numbers that is a solution to equation (1) is a solution to equation (2).
Based on the law of reversibility of equality, we swap the sides of equation (1). Pythagorean numbers (z, x, y) can be interpreted as side lengths right triangle, and squares (x 2 , y 2 , z 2) can be interpreted as the area of squares built on its hypotenuse and legs.
Let us multiply the areas of the squares of equation (1) by an arbitrary height h :
z 2 h = x 2 h + y 2 h (3)
Equation (3) can be interpreted as the equality of the volume of a parallelepiped to the sum of the volumes of two parallelepipeds.
Let the height of three parallelepipeds h = z :
z 3 = x 2 z + y 2 z (4)
The volume of the cube is decomposed into two volumes of two parallelepipeds. We will leave the volume of the cube unchanged, and reduce the height of the first parallelepiped to x and reduce the height of the second parallelepiped to y . The volume of a cube is greater than the sum of the volumes of two cubes:
z 3 > x 3 + y 3 (5)
On the set of triples of Pythagorean numbers ( x, y, z ) at n=3 there cannot be any solution to equation (2). Consequently, on the set of all triples of Pythagorean numbers it is impossible to decompose a cube into two cubes.
Let in equation (3) the height of three parallelepipeds h = z 2 :
z 2 z 2 = x 2 z 2 + y 2 z 2 (6)
The volume of a parallelepiped is decomposed into the sum of the volumes of two parallelepipeds.
We leave the left side of equation (6) unchanged. On its right side the height z 2
reduce to X
in the first term and before at 2
in the second term.
Equation (6) turned into inequality:
The volume of the parallelepiped is decomposed into two volumes of two parallelepipeds.
We leave the left side of equation (8) unchanged.
On the right side the height zn-2
reduce to xn-2
in the first term and reduce to y n-2
in the second term. Equation (8) becomes inequality:
| z n > x n + y n | (9) |
On the set of triplets of Pythagorean numbers there cannot be a single solution to equation (2).
Consequently, on the set of all triples of Pythagorean numbers for all n > 2 equation (2) has no solutions.
A “truly miraculous proof” has been obtained, but only for triplets Pythagorean numbers. This is lack of evidence and the reason for P. Fermat’s refusal from him.
B) Let us prove that equation (2) has no solutions on the set of triplets of non-Pythagorean numbers, which represents a family of an arbitrary triple of Pythagorean numbers z = 13, x = 12, y = 5 and a family of an arbitrary triple of positive integers z = 21, x = 19, y = 16
Both triplets of numbers are members of their families:
| (13, 12, 12); (13, 12,11);…; (13, 12, 5) ;…; (13,7, 1);…; (13,1, 1) | (10) | |
| (21, 20, 20); (21, 20, 19);…;(21, 19, 16);…;(21, 1, 1) | (11) |
The number of family members (10) and (11) is equal to half the product of 13 by 12 and 21 by 20, i.e. 78 and 210.
Each family member (10) contains z = 13 and variables X And at 13 > x > 0 , 13 > y > 0 1
Each member of the family (11) contains z = 21 and variables X And at , which take integer values 21 > x >0 , 21 > y > 0 . Variables successively decrease by 1 .
Triples of numbers of the sequence (10) and (11) can be represented as a sequence of inequalities of the third degree:
| 13 3 < 12 3 + 12 3 ;13 3 < 12 3 + 11 3 ;…; 13 3 < 12 3 + 8 3 ; 13 3 > 12 3 + 7 3 ;…; 13 3 > 1 3 + 1 3 | ||
| 21 3 < 20 3 + 20 3 ; 21 3 < 20 3 + 19 3 ; …; 21 3 < 19 3 + 14 3 ; 21 3 > 19 3 + 13 3 ;…; 21 3 > 1 3 + 1 3 |
and in the form of inequalities of the fourth degree:
| 13 4 < 12 4 + 12 4 ;…; 13 4 < 12 4 + 10 4 ; 13 4 > 12 4 + 9 4 ;…; 13 4 > 1 4 + 1 4 | ||
| 21 4 < 20 4 + 20 4 ; 21 4 < 20 4 + 19 4 ; …; 21 4 < 19 4 + 16 4 ;…; 21 4 > 1 4 + 1 4 |
The correctness of each inequality is verified by raising the numbers to the third and fourth powers.
A cube of a larger number cannot be decomposed into two cubes of smaller numbers. It is either less or greater than the sum of the cubes of the two smaller numbers.
The biquadratic of a larger number cannot be decomposed into two biquadrates of smaller numbers. It is either less or greater than the sum of the bisquares of smaller numbers.
As the exponent increases, all inequalities, except the left extreme inequality, have the same meaning:
They all have the same meaning: the power of the larger number is greater than the sum of the powers of the smaller two numbers with the same exponent:
| 13 n > 12 n + 12 n ; 13 n > 12 n + 11 n ;…; 13 n > 7 n + 4 n ;…; 13 n > 1 n + 1 n | (12) | |
| 21 n > 20 n + 20 n ; 21 n > 20 n + 19 n ;…; ;…; 21 n > 1 n + 1 n | (13) |
The left extreme term of sequences (12) (13) represents the weakest inequality. Its correctness determines the correctness of all subsequent inequalities of sequence (12) for n > 8 and sequence (13) at n > 14 .
There can be no equality among them. An arbitrary triple of positive integers (21,19,16) is not a solution to equation (2) of Fermat’s last theorem. If an arbitrary triple of positive integers is not a solution to the equation, then the equation has no solutions on the set of positive integers, which is what needed to be proved.
WITH) Fermat's commentary on Diophantus' problem states that it is impossible to decompose " in general, no power greater than a square, two powers with the same exponent».
Kiss a degree greater than a square cannot really be decomposed into two degrees with the same exponent. No kisses a degree greater than a square can be decomposed into two powers with the same exponent.
Any arbitrary triple of positive integers (z, x, y) may belong to a family, each member of which consists of a constant number z and two numbers smaller z . Each member of the family can be represented in the form of an inequality, and all resulting inequalities can be represented in the form of a sequence of inequalities:
| z n< (z — 1) n + (z — 1) n ; z n < (z — 1) n + (z — 2) n ; …; z n >1 n + 1 n | (14) |
The sequence of inequalities (14) begins with inequalities whose left side is smaller right side, but ends in inequalities in which the right side is smaller than the left side. With increasing exponent n > 2 the number of inequalities on the right side of sequence (14) increases. With the exponent n = k all inequalities on the left side of the sequence change their meaning and take on the meaning of the inequalities on the right side of the inequalities of sequence (14). As a result of increasing the exponent of all inequalities, the left side turns out to be larger than the right side:
| z k > (z-1) k + (z-1) k ; z k > (z-1) k + (z-2) k ;…; z k > 2 k + 1 k ; z k > 1 k + 1 k | (15) |
With a further increase in the exponent n>k none of the inequalities changes its meaning and turns into equality. On this basis, it can be argued that any arbitrarily chosen triple of positive integers (z, x, y) at n > 2 , z > x , z > y
In an arbitrarily chosen triple of positive integers z can be an arbitrarily large natural number. For all natural numbers, which are no more z , Fermat's Last Theorem is proven.
D) No matter how large the number z , in the natural series of numbers there is a large but finite set of integers before it, and after it there is an infinite set of integers.
Let us prove that the entire infinite set of natural numbers large z , form triples of numbers that are not solutions to the equation of Fermat’s Last Theorem, for example, an arbitrary triple of positive integers (z + 1, x ,y) , wherein z + 1 > x And z + 1 > y for all values of the exponent n > 2 is not a solution to the equation of Fermat's last theorem.
A randomly selected triple of positive integers (z + 1, x, y) may belong to a family of triples of numbers, each member of which consists of a constant number z+1 and two numbers X And at , taking on different values, smaller z+1 . Members of the family can be represented in the form of inequalities in which the constant left side is less than, or greater than, the right side. The inequalities can be ordered in the form of a sequence of inequalities:
With a further increase in the exponent n>k to infinity, none of the inequalities of sequence (17) changes its meaning and turns into equality. In sequence (16), the inequality formed from an arbitrarily chosen triple of positive integers (z + 1, x, y) , can be located on its right side in the form (z + 1) n > x n + y n or be on its left side in the form (z+1)n< x n + y n .
In any case, a triple of positive integers (z + 1, x, y) at n > 2 , z + 1 > x , z + 1 > y in sequence (16) represents an inequality and cannot represent an equality, that is, it cannot represent a solution to the equation of Fermat’s last theorem.
It is easy and simple to understand the origin of the sequence of power inequalities (16), in which the last inequality on the left side and the first inequality on the right side are inequalities of opposite meaning. On the contrary, it is not easy and difficult for schoolchildren, high school students and high school students, to understand how a sequence of inequalities (16) is formed from a sequence of inequalities (17), in which all inequalities have the same meaning.
In sequence (16), increasing the integer degree of inequalities by 1 unit turns the last inequality on the left side into the first inequality of the opposite sense on the right side. Thus, the number of inequalities on the left side of the sequence decreases, and the number of inequalities on the right side increases. Between the last and first power inequalities of opposite meaning in mandatory there is a power equality. Its degree cannot be an integer, since only non-integer numbers lie between two consecutive natural numbers. A power equality of a non-integer degree, according to the conditions of the theorem, cannot be considered a solution to equation (1).
If in sequence (16) we continue to increase the degree by 1 unit, then the last inequality of its left side will turn into the first inequality of the opposite meaning of the right side. As a result, there will be no left-hand inequalities left and only right-hand inequalities will remain, which will be a sequence of increasing power inequalities (17). A further increase in their integer power by 1 unit only strengthens its power inequalities and categorically excludes the possibility of equality in the integer power.
Consequently, in general, no integer power of a natural number (z+1) of the sequence of power inequalities (17) can be decomposed into two integer powers with the same exponent. Therefore, equation (1) has no solutions on an infinite set of natural numbers, which is what needed to be proved.
Consequently, Fermat's last theorem is proven in its entirety:
- in section A) for all triplets (z, x, y) Pythagorean numbers (Fermat’s discovery is truly a wonderful proof),
- in section B) for all members of the family of any triple (z, x, y) Pythagorean numbers,
- in section C) for all triples of numbers (z, x, y) , not large numbers z
- in section D) for all triples of numbers (z, x, y) natural series of numbers.
|
Changes made 09/05/2010 |
Which theorems can and cannot be proven by contradiction?
The explanatory dictionary of mathematical terms defines a proof by contradiction of a theorem, the opposite of a converse theorem.
“Proof by contradiction is a method of proving a theorem (proposition), which consists in proving not the theorem itself, but its equivalent (equivalent) theorem. Proof by contradiction is used whenever the direct theorem is difficult to prove, but the opposite theorem is easier to prove. In a proof by contradiction, the conclusion of the theorem is replaced by its negation, and through reasoning one arrives at the negation of the conditions, i.e. to a contradiction, to the opposite (the opposite of what is given; this reduction to the absurd proves the theorem."
Proof by contradiction is very often used in mathematics. Proof by contradiction is based on the law of excluded middle, which consists in the fact that of two statements (statements) A and A (negation of A), one of them is true and the other is false.”/Explanatory Dictionary of Mathematical Terms: A Manual for Teachers/O. V. Manturov [etc.]; edited by V. A. Ditkina.- M.: Education, 1965.- 539 p.: ill.-C.112/.
It would not be better to openly declare that the method of proof by contradiction is not a mathematical method, although it is used in mathematics, that it is a logical method and belongs to logic. Is it acceptable to say that proof by contradiction is “used whenever a direct theorem is difficult to prove,” when in fact it is used when, and only when, there is no substitute.
Deserves special attention and a characteristic of the relationship of the direct and inverse theorems to each other. “The converse theorem for a given theorem (or to a given theorem) is a theorem in which the condition is the conclusion, and the conclusion is the condition of the given theorem. This theorem in relation to the converse theorem is called the direct theorem (original). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and converse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If in a quadrilateral the diagonals are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e. the converse theorem is false.”/Explanatory Dictionary of Mathematical Terms: A Manual for Teachers/O. V. Manturov [etc.]; edited by V. A. Ditkina.- M.: Education, 1965.- 539 p.: ill.-C.261 /.
This characteristic The relationship between the direct and inverse theorems does not take into account the fact that the condition of the direct theorem is accepted as given, without proof, so its correctness is not guaranteed. The condition of the inverse theorem is not accepted as given, since it is the conclusion of the proven direct theorem. Its correctness is confirmed by the proof of the direct theorem. This essential logical difference in the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and cannot be proved by the logical method by contradiction.
Let us assume that there is a direct theorem in mind, which can be proven using the usual mathematical method, but is difficult. Let us formulate it in general view in short form like this: from A should E . Symbol A has the meaning of the given condition of the theorem, accepted without proof. Symbol E what matters is the conclusion of the theorem that needs to be proven.
We will prove the direct theorem by contradiction, logical method. The logical method is used to prove a theorem that has not mathematical condition, and logical condition. It can be obtained if the mathematical condition of the theorem from A should E , supplement with the exact opposite condition from A do not do it E .
The result was a logical contradictory condition of the new theorem, containing two parts: from A should E And from A do not do it E . The resulting condition of the new theorem corresponds to the logical law of excluded middle and corresponds to the proof of the theorem by contradiction.
According to the law, one part of a contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has the task and purpose of establishing exactly which part of the two parts of the condition of the theorem is false. Once the false part of the condition is determined, the other part is determined to be the true part, and the third is excluded.
According to the explanatory dictionary of mathematical terms, “proof is reasoning during which the truth or falsity of any statement (judgment, statement, theorem) is established”. Proof by contradiction there is a reasoning during which it is established falsity(absurdity) of the conclusion arising from false conditions of the theorem to be proved.
Given: from A should E and from A do not do it E .
Prove: from A should E .
Proof: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its resolution in the proof and its result. The result turns out to be false with flawless and error-free reasoning. The reason for a false conclusion in logically correct reasoning can only be a contradictory condition: from A should E And from A do not do it E .
There is no shadow of doubt that one part of the condition is false, and the other in this case is true. Both parts of the condition have the same origin, are accepted as data, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature was discovered that would distinguish one part of the condition from the other. Therefore, to the same extent it may be from A should E and maybe from A do not do it E . Statement from A should E May be false, then the statement from A do not do it E will be true. Statement from A do not do it E may be false, then the statement from A should E will be true.
Consequently, it is impossible to prove a direct theorem by contradiction.
Now we will prove this same direct theorem using the usual mathematical method.
Given: A .
Prove: from A should E .
Proof.
1. From A should B
2. From B should IN (according to the previously proven theorem)).
3. From IN should G (according to the previously proven theorem).
4. From G should D (according to the previously proven theorem).
5. From D should E (according to the previously proven theorem).
Based on the law of transitivity, from A should E . The direct theorem is proved by the usual method.
Let the proven direct theorem have a correct inverse theorem: from E should A .
Let's prove it with the usual mathematical method. The proof of the converse theorem can be expressed in symbolic form as an algorithm of mathematical operations.
Given: E
Prove: from E should A .
Proof.
1. From E should D
2. From D should G (according to the previously proven converse theorem).
3. From G should IN (according to the previously proven converse theorem).
4. From IN do not do it B (the converse theorem is not true). That's why from B do not do it A .
In this situation, it makes no sense to continue the mathematical proof of the converse theorem. The reason for the situation is logical. An incorrect converse theorem cannot be replaced by anything. Therefore, it is impossible to prove this converse theorem using the usual mathematical method. All hope is to prove this inverse theorem by contradiction.
In order to prove it by contradiction, it is necessary to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.
Converse theorem states: from E do not do it A . Her condition E , from which the conclusion follows A , is the result of proving the direct theorem using the usual mathematical method. This condition must be preserved and supplemented with the statement from E should A . As a result of the addition, we obtain the contradictory condition of the new inverse theorem: from E should A And from E do not do it A . Based on this logically contradictory condition, the converse theorem can be proven by means of the correct logical reasoning only, and only, logical method by contradiction. In a proof by contradiction, any mathematical actions and operations are subordinated to logical ones and therefore do not count.
In the first part of the contradictory statement from E should A condition E was proved by the proof of the direct theorem. In the second part from E do not do it A condition E was assumed and accepted without proof. One of them is false, and the other is true. You need to prove which one is false.
We prove it through correct logical reasoning and discover that its result is a false, absurd conclusion. The reason for a false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. The false part can only be a statement from E do not do it A , in which E was accepted without proof. This is what makes it different from E statements from E should A , which is proved by the proof of the direct theorem.
Therefore, the statement is true: from E should A , which was what needed to be proven.
Conclusion: by the logical method, only the inverse theorem is proven by contradiction, which has a direct theorem proven by the mathematical method and which cannot be proved by the mathematical method.
The obtained conclusion acquires exceptional importance in relation to the method of proof by contradiction of Fermat's great theorem. The overwhelming majority of attempts to prove it are based not on the usual mathematical method, but on the logical method of proof by contradiction. Wiles's proof of Fermat's Last Theorem is no exception.
Dmitry Abrarov, in the article “Fermat’s Theorem: the Phenomenon of Wiles’ Proofs,” published a commentary on Wiles’s proof of Fermat’s Last Theorem. According to Abrarov, Wiles proves Fermat's last theorem with the help of a remarkable discovery by the German mathematician Gerhard Frey (b. 1944), who related the potential solution of Fermat's equation x n + y n = z n
, Where n > 2
, with another, completely different equation. This new equation is given by a special curve (called Frey's elliptic curve). The Frey curve is given by a very simple equation:
.
“It was Frey who compared to every decision (a, b, c) Fermat's equation, that is, numbers satisfying the relation a n + b n = c n, the above curve. In this case, Fermat’s last theorem would follow.”(Quote from: Abrarov D. “Fermat’s Theorem: the phenomenon of Wiles’ proofs”)
In other words, Gerhard Frey suggested that the equation of Fermat's Last Theorem x n + y n = z n
, Where n > 2
, has solutions in positive integers. These same solutions are, according to Frey’s assumption, solutions to his equation
y 2 + x (x - a n) (y + b n) = 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable discovery by Frey and, with its help, mathematical method proved that this find, that is, the Frey elliptic curve, does not exist. Therefore, there is no equation and its solutions that are given by a non-existent elliptic curve. Therefore, Wiles should have accepted the conclusion that there is no equation of Fermat's last theorem and Fermat's theorem itself. However, he accepts a more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.
An irrefutable fact may be that Wiles accepted an assumption that is exactly the opposite in meaning to what is stated by Fermat’s great theorem. It obliges Wiles to prove Fermat's last theorem by contradiction. Let us follow his example and see what comes of this example.
Fermat's Last Theorem states that the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers.
According to the logical method of proof by contradiction, this statement is retained, accepted as given without proof, and then supplemented with an opposite statement: equation x n + y n = z n , Where n > 2 , has solutions in positive integers.
The presumptive statement is also accepted as given, without proof. Both statements, considered from the point of view of the basic laws of logic, are equally valid, equally valid and equally possible. Through correct reasoning, it is necessary to determine which one is false in order to then determine that the other statement is true.
Correct reasoning ends in a false, absurd conclusion, the logical reason for which can only be the contradictory condition of the theorem being proven, which contains two parts of directly opposite meaning. They were the logical reason for the absurd conclusion, the result of proof by contradiction.
However, in the course of logically correct reasoning, not a single sign was discovered by which it could be established which particular statement is false. It could be a statement: equation x n + y n = z n , Where n > 2 , has solutions in positive integers. On the same basis, it could be the following statement: equation x n + y n = z n , Where n > 2 , has no solutions in positive integers.
As a result of the reasoning, there can be only one conclusion: Fermat's Last Theorem cannot be proven by contradiction.
It would be a completely different matter if Fermat's last theorem were an inverse theorem, which has a direct theorem proven by the usual mathematical method. In this case, it could be proven by contradiction. And since it is a direct theorem, its proof should be based not on the logical method of proof by contradiction, but on the ordinary mathematical method.
According to D. Abrarov, the most famous of modern Russian mathematicians, Academician V. I. Arnold, reacted “actively skeptically” to Wiles’ proof. The academician stated: “this is not real mathematics - real mathematics is geometric and has strong connections with physics.” (Quote from: Abrarov D. “Fermat’s Theorem: the phenomenon of Wiles’ proofs.” The academician’s statement expresses the very essence of Wiles’ non-mathematical proof of Fermat’s last theorem.
By contradiction it is impossible to prove either that the equation of Fermat's Last Theorem has no solutions or that it has solutions. Wiles' mistake is not mathematical, but logical - the use of proof by contradiction where its use does not make sense and Fermat's great theorem does not prove.
Fermat's Last Theorem cannot be proved using the usual mathematical method, if it contains: equation x n + y n = z n , Where n > 2 , has no solutions in positive integers, and if you want to prove in it: the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers. In this form there is not a theorem, but a tautology devoid of meaning.
Note. My BTF proof was discussed on one of the forums. One of Trotil's members, an expert in number theory, made the following authoritative statement entitled: " Brief retelling what Mirgorodsky did." I quote it verbatim:
« A. He proved that if z 2 = x 2 + y , That z n > x n + y n . This is a well-known and quite obvious fact.
IN. He took two triples - Pythagorean and non-Pythagorean and showed by simple search that for a specific, specific family of triples (78 and 210 pieces) the BTF is satisfied (and only for it).
WITH. And then the author omitted the fact that from < to a later extent it may turn out to be = , not only > . A simple counterexample - transition n=1 V n=2 in the Pythagorean triple.
D. This point does not contribute anything significant to the BTF proof. Conclusion: BTF has not been proven.”
I will consider his conclusion point by point.
A. It proves the BTF for the entire infinite set of triples of Pythagorean numbers. Proved by a geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was discovered, as I believe, by P. Fermat himself. Fermat may have had this in mind when he wrote:
“I have discovered a truly wonderful proof of this, but these fields are too narrow for it.” This assumption of mine is based on the fact that in the Diophantine problem, against which Fermat wrote in the margins of the book, we are talking about solutions to the Diophantine equation, which are triplets of Pythagorean numbers.
An infinite set of triplets of Pythagorean numbers are solutions to the Diophatean equation, and in Fermat’s theorem, on the contrary, none of the solutions can be a solution to the equation of Fermat’s theorem. And Fermat’s truly wonderful proof is directly related to this fact. Fermat could later extend his theorem to the set of all natural numbers. On the set of all natural numbers, BTF does not belong to the “set of exceptionally beautiful theorems.” This is my assumption, which can neither be proven nor disproved. It can be accepted or rejected.
IN. At this point, I prove that both the family of an arbitrarily taken Pythagorean triple of numbers and the family of an arbitrarily taken non-Pythagorean triple of BTF numbers are satisfied. This is a necessary, but insufficient and intermediate link in my proof of BTF. The examples I took of the family of the triple of Pythagorean numbers and the family of the triple of non-Pythagorean numbers have the meaning of specific examples that presuppose and do not exclude the existence of similar other examples.
Trotil’s statement that I “showed by simple search that for a specific, specific family of triplets (78 and 210 pieces) the BTF is satisfied (and only for it) is baseless. He cannot refute the fact that I can just as easily take other examples of Pythagorean and non-Pythagorean triples to obtain a specific definite family of one and the other triple.
Whatever pair of triplets I take, checking their suitability for solving the problem can be carried out, in my opinion, only by the “simple enumeration” method. I don't know any other method and don't need it. If Trotil didn't like it, then he should have suggested another method, which he doesn't do. Without offering anything in return, it is incorrect to condemn “simple overkill,” which in this case is irreplaceable.
WITH. I have omitted = between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 = x 2 + y (1), in which the degree n > 2 — whole positive number. From the equality between the inequalities it follows mandatory consideration of equation (1) for a non-integer degree value n > 2 . Trotil, counting compulsory consideration of equality between inequalities actually considers necessary in the BTF proof, consideration of equation (1) with not whole degree value n > 2 . I did this for myself and found that equation (1) with not whole degree value n > 2 has a solution of three numbers: z, (z-1), (z-1) for a non-integer exponent.
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.
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 a 5th grade level. high school, but the proof is not even for every professional mathematician. 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.
And so on. What if we take a similar equation x³+y³=z³? Maybe there are such numbers too?
And so on (Fig. 1).
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’s 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 explored 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 also prime numbers infinitely many...
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 better to do, he went to the library and began to read famous article Kummera. 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 conjecture. 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 contains a gross error, although in general it is 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... Fermat's interest in mathematics appeared unexpectedly and quite unexpectedly. mature age. In 1629, a Latin translation of Pappus's work, containing a brief summary of Apollonius' results on the properties of conic sections, fell into his hands. Fermat, a polyglot, an expert in law and ancient philology, suddenly sets out to completely restore the course of reasoning of the famous scientist. With the same success, a modern lawyer can try to independently reproduce all the evidence from a monograph from problems, say, algebraic topology. However, the unthinkable undertaking is crowned with success. Moreover, delving into the geometric constructions of the ancients, he makes amazing discovery: To find the maximum and minimum areas of figures, you don’t need fancy drawings. It is always possible to construct and solve some simple algebraic equation, the roots of which determine the extremum. He came up with an algorithm that would become the basis of differential calculus.
He quickly moved on. He found sufficient conditions for the existence of maxima, learned to determine inflection points, and drew tangents to all known second- and third-order curves. A few more years and he finds a new one algebraic method finding quadratures for parabolas and hyperbolas of arbitrary order (that is, integrals of functions of the form y p = Cx q And y p x q = C), calculates areas, volumes, moments of inertia of bodies of revolution. It was a real breakthrough. Feeling this, Fermat begins to seek communication with the mathematical authorities of the time. He is confident and craves recognition.
In 1636, he wrote his first letter to His Reverend Marin Mersenne: “Holy Father! I am extremely grateful to you for the honor that you have shown me by giving me hope that we will be able to talk in writing; ...I will be very glad to learn from you about all the new treatises and books on Mathematics that have appeared over the past five or six years. ...I have also found many analytical methods for various problems, both numerical and geometric, for the solution of which Vieta's analysis is insufficient. I will share all this with you whenever you want, and without any arrogance, from which I am freer and more distant than any other person in the world.”
Who is Father Mersenne? This is a Franciscan monk, a scientist of modest talents and a remarkable organizer, who for 30 years headed the Parisian mathematical circle, which became the true center of French science. Subsequently, the Mersenne circle, by decree of Louis XIV, would be transformed into the Paris Academy of Sciences. Mersenne tirelessly carried on a huge correspondence, and his cell in the monastery of the Order of Minims on the Royal Square was a kind of “post office for all the scientists of Europe, from Galileo to Hobbes.” Correspondence then replaced scientific journals, which appeared much later. Meetings at Mersenne's took place weekly. The core of the circle consisted of the most brilliant naturalists of that time: Robertville, Pascal the Father, Desargues, Midorge, Hardy and, of course, the famous and universally recognized Descartes. René du Perron Descartes (Cartesius), nobleman's mantle, two family estates, founder of Cartesianism, “father” of analytical geometry, one of the founders of new mathematics, as well as Mersenne’s friend and fellow student at the Jesuit college. This wonderful person will be a nightmare for Fermat.
Mersenne found Fermat's results interesting enough to introduce the provincial to his elite club. The farm immediately began correspondence with many members of the circle and was literally bombarded with letters from Mersenne himself. In addition, he sends completed manuscripts to the judgment of learned men: “Introduction to flat and solid places”, and a year later - “Method of finding maxima and minima” and “Answers to questions of B. Cavalieri”. What Fermat expounded was absolutely new, but there was no sensation. Contemporaries did not shudder. They understood little, but they found clear indications that Fermat borrowed the idea of the maximization algorithm from Johannes Kepler’s treatise with the amusing title “The New Stereometry of Wine Barrels.” Indeed, in Kepler’s reasoning there are phrases like “The volume of a figure is greatest if on both sides of the place of greatest value the decrease is at first insensitive.” But the idea of a small increment of a function near an extremum was not at all in the air. The best analytical minds of that time were not ready to manipulate small quantities. The fact is that at that time algebra was considered a kind of arithmetic, that is, second-class mathematics, a primitive tool at hand, developed for the needs of base practice (“only merchants count well”). Tradition prescribed adherence to purely geometric methods of proof, dating back to ancient mathematics. Fermat was the first to realize that infinitesimal quantities can be added and reduced, but it is quite difficult to represent them in the form of segments.
It took almost a century for Jean d'Alembert to admit in his famous Encyclopedia: “Fermat was the inventor of new calculus. It is with him that we find the first application of differentials to find tangents.” At the end of the 18th century, Joseph Louis Comte de Lagrange spoke out even more clearly: “But the geometers - Fermat’s contemporaries - did not understand this new kind of calculus. They saw only special cases. And this invention, which appeared shortly before Descartes’ Geometry, remained fruitless for forty years.” Lagrange is referring to 1674, when Isaac Barrow's Lectures were published, covering Fermat's method in detail.
Among other things, it quickly became clear that Fermat was more inclined to formulate new problems than to humbly solve the problems proposed by the meters. In the era of duels, the exchange of tasks between pundits was generally accepted as a form of clarifying problems associated with subordination. However, Fermat clearly does not know the limits. Each of his letters is a challenge containing dozens of complex unsolved problems, and on the most unexpected topics. Here is an example of his style (addressed to Frenicle de Bessy): “Item, what is least square, which, when reduced by 109 and added by one, will give a square? If you don't send me general solution, then send the quotient for these two numbers, which I chose small so as not to confuse you too much. After I receive your response, I will suggest some other things to you. It is clear without any special reservations that my proposal requires finding integers, since in the case fractional numbers the most insignificant arithmetician could arrive at the goal.” Fermat often repeated himself, formulating the same questions several times, and openly bluffed, claiming that he had an unusually elegant solution to the proposed problem. There were some direct mistakes too. Some of them were noticed by contemporaries, and some insidious statements misled readers for centuries.
The Mersenne circle reacted adequately. Only Robertville, the only member of the circle who had problems with his origin, maintains the friendly tone of the letters. The good shepherd Father Mersenne tried to reason with the “impudent Toulouse.” But Fermat does not intend to make excuses: “Reverend Father! You write to me that the posing of my impossible problems angered and cooled Messrs. Saint-Martin and Frenicle and that this was the reason for the cessation of their letters. However, I want to object to them that what seems at first impossible is not really so and that there are many problems that, as Archimedes said ... ”, etc..
However, Fermat is disingenuous. It was to Frenicles that he sent the problem of finding a right triangle with integer sides, the area of which is equal to the square of the integer. I sent it, although I knew that the problem obviously had no solution.
Descartes took the most hostile position towards Fermat. In his letter to Mersenne from 1938 we read: “since I learned that this is the same man who had previously tried to refute my Dioptrics, and since you informed me that he sent this after reading my Geometry ” and in surprise that I did not find the same thing, that is, (as I have reason to interpret it) sent it with the aim of entering into rivalry and showing that in this he knows more than I, and since even of yours letters, I learned that he has a reputation as a very knowledgeable geometer, then I consider myself obliged to answer him.” Descartes would later solemnly designate his answer as “the small process of Mathematics against Mr. Fermat.”
It is easy to understand what infuriated the eminent scientist. Firstly, Fermat’s arguments constantly include coordinate axes and the representation of numbers by segments - a technique that Descartes comprehensively develops in his just published Geometry. Fermat comes to the idea of replacing drawings with calculations completely independently; in some ways he is even more consistent than Descartes. Secondly, Fermat brilliantly demonstrates the effectiveness of his method of finding minima using the example of the problem of the shortest path of a light ray, clarifying and supplementing Descartes with his “Dioptrics”.
The merits of Descartes as a thinker and innovator are enormous, but let’s open the modern “Mathematical Encyclopedia” and look at the list of terms associated with his name: “Cartesian coordinates” (Leibniz, 1692), “Cartesian sheet”, “Cartesian ovals”. None of his arguments went down in history as “Descartes’ Theorem.” Descartes is first and foremost an ideologist: he is the founder of a philosophical school, he forms concepts, improves the system of letter symbols, but his creative heritage contains few new specific techniques. In contrast, Pierre Fermat writes little, but for any reason he can come up with a lot of ingenious mathematical tricks (see also “Fermat’s Theorem”, “Fermat’s Principle”, “Fermat’s Method of Infinite Descent”). They were probably quite rightly jealous of each other. A collision was inevitable. With the Jesuit mediation of Mersenne, a war broke out that lasted two years. However, Mersenne turned out to be right here before history: the fierce battle of the two titans, their intense, to put it mildly, polemics contributed to the understanding of the key concepts of mathematical analysis.
Fermat is the first to lose interest in the discussion. Apparently, he explained himself directly to Descartes and never again offended his opponent. In one of their latest works“Synthesis for Refraction,” the manuscript of which he sent to de la Chambre, Fermat through the word remembers “the most learned Descartes” and in every possible way emphasizes his priority in matters of optics. Meanwhile, it was this manuscript that contained a description of the famous “Fermat's principle,” which provides a comprehensive explanation of the laws of reflection and refraction of light. Nods to Descartes in work of this level were completely unnecessary.
What happened? Why did Fermat, putting aside his pride, go for reconciliation? Reading Fermat's letters of those years (1638 - 1640), one can assume the simplest thing: during this period his scientific interests changed dramatically. He abandons the fashionable cycloid, ceases to be interested in tangents and areas, and for many 20 years forgets about his method of finding the maximum. Having enormous merits in the mathematics of the continuous, Fermat completely immersed himself in the mathematics of the discrete, leaving disgusting geometric drawings to his opponents. Numbers become his new passion. As a matter of fact, the entire “Number Theory”, as an independent mathematical discipline, owes its birth entirely to the life and work of Fermat.
<…>After Fermat’s death, his son Samuel published in 1670 a copy of “Arithmetic” belonging to his father under the title “Six books of arithmetic by the Alexandrian Diophantus with comments by L. G. Bachet and remarks by P. de Fermat, Toulouse senator.” The book also included some letters from Descartes and full text the works of Jacques de Bigly “A New Discovery in the Art of Analysis”, written on the basis of Fermat’s letters. The publication was an incredible success. An unprecedented bright world opened up before the amazed specialists. The unexpectedness, and most importantly the accessibility, democracy of Fermat’s number-theoretic results gave rise to a lot of imitations. At that time, few people understood how the area of a parabola is calculated, but every student could understand the formulation of Fermat's Last Theorem. A real hunt began for the scientist's unknown and lost letters. Until the end of the 17th century. Every word of his found was published and republished. But the turbulent history of the development of Fermat’s ideas was just beginning.
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, the Clay Mathematics Institute offered a prize to those who solved problems known to be the hardest unsolvable problems, calling its list the 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 of quantum Yang-Mills theory;
- 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. Her proof is in general case hasn't 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 understandable to the average 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 must become a tool to explain all these interactions. Yang-Mills theory is a mathematical language with which it has become 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 the opposite direction, that is, 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 its set rational decisions are related to the behavior of the L-function at unity. 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 domestic digital signature standards are based on their use.
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 classes p and np, also known as the Cook-Lewin problem, in clear language can be formulated 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, it will be necessary to reconsider many fundamental principles modern cryptography, which form the basis of a significant part of electronic commerce mechanisms.
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 that it is possible to approximate the shape of any object by "gluing" it together simple bodies larger 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 reach new round technological development.
Sometimes diligent study of the exact sciences can bear fruit - you will become not only famous throughout the 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 for example the Kourovsky or Dniester notebooks, sort of collections with unsolvable physical and mathematical problems, and not only, tasks. However, there are also truly complex theorems that have not been solved for decades, and for them the American Clay Institute has awarded a reward of $1 million for each. Until 2002, the total jackpot was 7 million, since there were seven “Millennium Problems,” but Russian mathematician Grigory Perelman solved the Poincaré conjecture by epically giving up a million without even opening the door to US mathematicians who wanted to give him his hard-earned bonus. So, let's turn on The Big Bang Theory for background and mood, and see what else you can make a tidy sum of money for.
Equality of classes P and NP
In simple terms, the problem of equality P = NP is the following: if the positive answer to some question can be checked quite quickly (in polynomial time), then is it true that the answer to this question can be found quite quickly (also in polynomial time and using polynomial memory)? In other words, is it really not easier to check the solution to a problem than to find it? The point here is that some calculations and calculations are easier to solve using an algorithm rather than brute force, and thus save a lot of time and resources.
Hodge conjecture
Hodge's hypothesis was formulated in 1941 and states that for especially good types 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 a geometric essence, “sort of scary multi-dimensional scribbles,” or you can still get by with conditionally standard algebra + geometry.
Riemann hypothesis
It is quite difficult to explain in human language; it is enough to know that the solution to this problem will have far-reaching consequences in the field of distribution of prime numbers. The problem is so important and pressing that even deducing a counterexample to the hypothesis - at the discretion of the academic council of the university, the problem can be considered proven, so here you can try the “reverse” method. Even if it is possible to reformulate the hypothesis into a more in the narrow sense- and then the Clay Institute will pay 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 guys 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 modeling, but no one can prove it yet.
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 movements of a viscous Newtonian fluid; they are of great practical importance, and most importantly they describe turbulence, which cannot be driven into the framework of science and its properties and actions cannot be predicted. Justification for the construction of these equations would allow us not to point our fingers at the sky, but to understand turbulence from the inside and make planes and mechanisms more stable.
Birch-Swinnerton-Dyer conjecture
Here, however, I tried to find simple words, but there is such dense algebra here that it is impossible to do without a deep dive. Those who do not want to scuba dive into the matan should 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 proving this hypothesis will enrich you by a million dollars.
It should be noted that there has already been progress in almost every area, and even cases have been proven for individual examples. Therefore, you should 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).