The History of Farm's Great Theorem. Felix Kirsanov

Pierre de 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 remarks 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 neither a cube into two cubes, nor a bi-square into two bi-squares, and, in general, no degree greater than a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, but these margins are too narrow for it.» / E.T.Bell "Creators of Mathematics". M., 1979, p.69/. I bring to your attention an elementary proof of the farm theorem, which can be understood by any high school student who is fond of mathematics.

Let us compare Fermat's commentary on the Diophantine problem with the modern formulation of Fermat's great 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 affirmed by the problem of Diophantus, on the contrary, is affirmed by Fermat's commentary.

Fermat's comment can be interpreted as follows: if a quadratic equation with three unknowns has an infinite number of solutions on the set of all triples Pythagorean numbers, then, conversely, an equation with three unknowns to a power greater than the square

There is not even a hint of its connection with the Diophantine problem in the equation. His assertion requires proof, but it does not have a condition from which it follows that it has no solutions in positive integers.

The variants of the proof of the equation known to me are reduced to the following algorithm.

1. The equation of Fermat's theorem is taken as its conclusion, the validity of which is verified with the help of proof.
2. The same equation is called original the equation from which its proof must proceed.

The result is a tautology: If an equation has no solutions in positive integers, then it has no solutions in positive integers.". The proof of the tautology is obviously wrong and devoid of any meaning. But it is proved by contradiction.

• An assumption is made that is the opposite of that stated by the equation to be proven. It should not contradict the original equation, but it does. To prove what is accepted without proof, and to accept without proof what is required to be proved, does not make sense.
• 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, the proof of the equation of Fermat's Last Theorem has remained an impossible dream of specialists and lovers of mathematics.

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 set 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.

BUT) Everyone knows that equation (1) has an infinite number of solutions on the set of all triples of Pythagorean numbers. Let us prove that no triple of Pythagorean numbers, which is a solution to equation (1), is a solution to equation (2).

Based on the law of reversibility of equality, the sides of equation (1) are interchanged. Pythagorean numbers (z, x, y) can be interpreted as the lengths of the sides of a right triangle, and the squares (x2, y2, z2) can be interpreted as the areas of squares built on its hypotenuse and legs.

We multiply 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 leave the volume of the cube unchanged, and reduce the height of the first parallelepiped to x and the height of the second parallelepiped will be reduced 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 can be no 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 = z2 :

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 z2 reduce to X in the first term and up to at 2 in the second term.

Equation (6) turned into the inequality:

The volume of a parallelepiped is decomposed into two volumes of two parallelepipeds.

We leave the left side of equation (8) unchanged.
On the right side of the height zn-2 reduce to xn-2 in the first term and reduce to y n-2 in the second term. Equation (8) turns into the inequality:

On the set of triples of Pythagorean numbers, there cannot be a single solution of equation (2).

Consequently, on the set of all triples of Pythagorean numbers for all n > 2 equation (2) has no solutions.

Obtained "post miraculous proof", but only for triplets Pythagorean numbers. This is lack of evidence and the reason for the refusal of P. Fermat from him.

b) Let us prove that equation (2) has no solutions on the set of triples of non-Pythagorean numbers, which is the family of an arbitrarily taken triple of Pythagorean numbers z=13, x=12, y=5 and the family of an arbitrary triple of positive integers z=21, x=19, y=16

Both triplets of numbers are members of their families:

The number of members of the family (10) and (11) is equal to half the product of 13 by 12 and 21 by 20, i.e. 78 and 210.

Each member of the family (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 . The variables decrease sequentially by 1 .

The triples of numbers of the sequence (10) and (11) can be represented as a sequence of inequalities of the third degree:

and in the form of inequalities of the fourth degree:

The correctness of each inequality is verified by raising the numbers to the third and fourth powers.

The cube of a larger number cannot be decomposed into two cubes of smaller numbers. It is either less than or greater than the sum of the cubes of the two smaller numbers.

The bi-square of a larger number cannot be decomposed into two bi-squares of smaller numbers. It is either less than or greater than the sum of the bi-squares of smaller numbers.

As the exponent increases, all inequalities, except for the leftmost inequality, have the same meaning:

Inequalities, they all have the same meaning: the degree of the larger number is greater than the sum of the degrees of the smaller two numbers with the same exponent:

The leftmost term of sequences (12) (13) is the weakest inequality. Its correctness determines the correctness of all subsequent inequalities of the sequence (12) for n > 8 and sequence (13) for 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 was to be proved.

FROM) Fermat's commentary on the Diophantus problem states that it is impossible to decompose " in general, no power greater than the square, two powers with the same exponent».

Kisses a power greater than a square cannot really be decomposed into two powers with the same exponent. I don't kiss a power greater than the square can be decomposed into two powers with the same exponent.

Any randomly chosen 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 less than z . Each member of the family can be represented in the form of an inequality, and all the resulting inequalities can be represented as a sequence of inequalities:

The sequence of inequalities (14) begins with inequalities whose left side is less than right side, but ends with inequalities whose right side is less than the left side. With increasing exponent n > 2 the number of inequalities on the right side of sequence (14) increases. With an exponent n=k all the inequalities of the left side of the sequence change their meaning and take on the meaning of the inequalities of the right side of the inequalities of the sequence (14). As a result of the increase in the exponent of all inequalities, the left side is greater than the right side:

With a further increase in the exponent n>k none of the inequalities changes its meaning and does not turn into equality. On this basis, it can be argued that any arbitrarily taken triple of positive integers (z, x, y) at n > 2 , z > x , z > y

In an arbitrary triple of positive integers z can be an arbitrarily large natural number. For all natural numbers not greater than z , Fermat's Last Theorem is proved.

D) No matter how big the number z , in the natural series of numbers before it there is a large but finite set of integers, and after it there is an infinite set of integers.

Let us prove that the entire infinite set of natural numbers greater than 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 chosen 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 different values, smaller z + 1 . Family members can be represented as inequalities whose constant left side is less than, or greater than, the right side. The inequalities can be arranged in order as a sequence of inequalities:

With a further increase in the exponent n>k to infinity, none of the inequalities in the sequence (17) changes its meaning and does not become an equality. In sequence (16), the inequality formed from an arbitrarily taken triple of positive integers (z + 1, x, y) , can be in 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, the triple of positive integers (z + 1, x, y) at n > 2 , z + 1 > x , z + 1 > y in sequence (16) is an inequality and cannot be an equality, i.e., it cannot be 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 of the left side and the first inequality of the right side are inequalities of the opposite sense. 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 (17) is formed from a sequence of inequalities (16), in which all inequalities have the same meaning.

In sequence (16), increasing the integer degree of inequalities by 1 turns the last inequality on the left side into the first inequality of the opposite meaning on the right side. Thus, the number of inequalities on the ninth side of the sequence decreases, while the number of inequalities on the right side increases. Between the last and first power inequalities of the opposite meaning, there is a power equality without fail. Its degree cannot be an integer, since there are only non-integer numbers between two consecutive natural numbers. The power equality of a non-integer degree, according to the condition of the theorem, cannot be considered a solution to equation (1).

If in the 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 inequalities on the left side and only inequalities on the right side, which will be a sequence of increasing power inequalities (17). A further increase in their integer degree by 1 unit only strengthens its power inequalities and categorically excludes the possibility of equality in an integer degree.

Therefore, 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 was to be proved.

Therefore, Fermat's Last Theorem is proved in all generality:

• in section A) for all triplets (z, x, y) Pythagorean numbers (Fermat's discovery is a truly miraculous proof),
• in section C) for all members of the family of any triple (z, x, y) pythagorean numbers,
• in section C) for all triplets of numbers (z, x, y) , not large numbers z
• in section D) for all triples of numbers (z, x, y) natural series of numbers.

Which theorems can and which cannot be proven by contradiction

The Explanatory Dictionary of Mathematical Terms defines proof by contradiction of a theorem opposite to the inverse theorem.

“Proof by contradiction is a method of proving a theorem (sentence), which consists in proving not the theorem itself, but its equivalent (equivalent), opposite inverse (reverse to opposite) theorem. Proof by contradiction is used whenever the direct theorem is difficult to prove, but the opposite inverse is easier. When proving by contradiction, the conclusion of the theorem is replaced by its negation, and by reasoning one arrives at the negation of the condition, i.e. to a contradiction, to the opposite (the opposite of what is given; this reduction to absurdity proves the theorem.

Proof by contradiction is very often used in mathematics. The proof by contradiction is based on the law of the excluded middle, which consists in the fact that of the 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 guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.112/.

It would not be better to declare openly 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 valid to say that proof by contradiction is "used whenever a direct theorem is difficult to prove", when in fact it is used if, and only if, there is no substitute for it.

The characteristic of the relationship between the direct and inverse theorems also deserves special attention. “An inverse 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 (initial). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and inverse 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 the diagonals in a quadrilateral are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e., the converse theorem is not true./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.261 /.

This characterization of the relationship between direct and inverse theorems does not take into account the fact that the condition of the direct theorem is taken as given, without proof, so that its correctness is not guaranteed. The condition of the inverse theorem is not taken 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 between the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and which cannot be proved by the logical method from the contrary.

Let's assume that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. Let's formulate it in general view in short form like this: from BUT should E . Symbol BUT has the value of the given condition of the theorem, accepted without proof. Symbol E is the conclusion of the theorem to be proved.

We will prove the direct theorem by contradiction, logical method. The logical method proves a theorem that has not mathematical condition, and logical condition. It can be obtained if the mathematical condition of the theorem from BUT should E , supplement with the opposite condition from BUT do not do it E .

As a result, a logical contradictory condition of the new theorem was obtained, which includes two parts: from BUT should E And from BUT do not do it E . The resulting condition of the new theorem corresponds to the logical law of the excluded middle and corresponds to the proof of the theorem by contradiction.

According to the law, one part of the contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has its own task and goal to establish exactly which part of the two parts of the condition of the theorem is false. As soon as the false part of the condition is determined, it will be established that the other part is 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 contrary there is a discussion in the course of which it is established falsity(absurdity) of the conclusion that follows from false conditions of the theorem being proved.

Given: from BUT should E and from BUT do not do it E .

Prove: from BUT 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 if the reasoning is flawless and infallible. The reason for a false conclusion with logically correct reasoning can only be a contradictory condition: from BUT should E And from BUT do not do it E .

There is no shadow of a 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 given, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature has been found that would distinguish one part of the condition from the other. Therefore, to the same extent, from BUT should E and maybe from BUT do not do it E . Statement from BUT should E may be false, then the statement from BUT do not do it E will be true. Statement from BUT do not do it E may be false, then the statement from BUT should E will be true.

Therefore, it is impossible to prove the direct theorem by contradiction method.

Now we will prove the same direct theorem by the usual mathematical method.

Given: BUT .

Prove: from BUT should E .

Proof.

1. From BUT should B

2. From B should IN (according to the previously proved theorem)).

3. From IN should G (according to the previously proved theorem).

4. From G should D (according to the previously proved theorem).

5. From D should E (according to the previously proved theorem).

Based on the law of transitivity, from BUT should E . The direct theorem is proved by the usual method.

Let the proven direct theorem have a correct converse theorem: from E should BUT .

Let's prove it by ordinary mathematical method. The proof of the inverse theorem can be expressed in symbolic form as an algorithm of mathematical operations.

Given: E

Prove: from E should BUT .

Proof.

1. From E should D

2. From D should G (by the previously proved inverse theorem).

3. From G should IN (by the previously proved inverse theorem).

4. From IN do not do it B (the converse is not true). That's why from B do not do it BUT .

In this situation, it makes no sense to continue the mathematical proof of the inverse theorem. The reason for the situation is logical. It is impossible to replace an incorrect inverse theorem with anything. Therefore, this inverse theorem cannot be proved by the usual mathematical method. All hope is to prove this inverse theorem by contradiction.

In order to prove it by contradiction, it is required to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.

Inverse theorem claims: from E do not do it BUT . Her condition E , from which follows the conclusion BUT , is the result of proving the direct theorem by the usual mathematical method. This condition must be retained and supplemented with the statement from E should BUT . As a result of the addition, a contradictory condition of the new inverse theorem is obtained: from E should BUT And from E do not do it BUT . Based on this logically contradictory condition, the converse theorem can be proved by the correct logical reasoning only, and only, logical opposite method. In a proof by contradiction, any mathematical actions and operations are subordinate to logical ones and therefore do not count.

In the first part of the contradictory statement from E should BUT condition E was proved by the proof of the direct theorem. In the second part from E do not do it BUT condition E was assumed and accepted without proof. One of them is false and the other is true. It is required to prove which of them is false.

We prove with the correct logical reasoning and find 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 BUT , in which E accepted without proof. This is what distinguishes it from E statements from E should BUT , which is proved by the proof of the direct theorem.

Therefore, the statement is true: from E should BUT , which was to be proved.

Output: only that converse theorem is proved by the logical method from the contrary, which has a direct theorem proved by the mathematical method and which cannot be proved by the mathematical method.

The conclusion obtained acquires an 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 proving by contradiction. The proof of Fermat Wiles' Great Theorem is no exception.

Dmitry Abrarov in his article "Fermat's Theorem: the Phenomenon of Wiles' Proofs" published a commentary on the proof of Fermat's Last Theorem by Wiles. According to Abrarov, Wiles proves Fermat's Last Theorem with the help of a remarkable finding by the German mathematician Gerhard Frey (b. 1944) relating a potential solution to 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 the Frey elliptic curve). The Frey curve is given by a very simple equation:
.

“It was precisely Frey who compared to every solution (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 proof")

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. The same solutions are, by Frey's assumption, the solutions of 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 of Frey and, with its help, through mathematical method proved that this finding, that is, Frey's 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 concluded that there is no equation of Fermat's Last Theorem and Fermat's Theorem itself. However, he takes the more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.

It may be an undeniable fact that Wiles accepted an assumption that is directly opposite in meaning to what is stated by Fermat's Last Theorem. It obliges Wiles to prove Fermat's Last Theorem by contradiction. Let's follow his example and see what happens from 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 preserved, accepted as given without proof, and then supplemented with a statement opposite in meaning: the equation x n + y n = z n , where n > 2 , has solutions in positive integers.

The hypothesized statement is also accepted as given, without proof. Both statements, considered from the point of view of the basic laws of logic, are equally admissible, equal in rights and equally possible. By correct reasoning, it is required to establish which of them is false, in order to then establish that the other statement is true.

Correct reasoning ends with a false, absurd conclusion, the logical cause of which can only be a contradictory condition of the theorem being proved, which contains two parts of a directly opposite meaning. They were the logical cause of the absurd conclusion, the result of proof by contradiction.

However, in the course of logically correct reasoning, not a single sign was found by which it would be possible to establish which particular statement is false. It can be a statement: the equation x n + y n = z n , where n > 2 , has solutions in positive integers. On the same basis, it can be the statement: the 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 very different matter if Fermat's Last Theorem were an inverse theorem that has a direct theorem proved by the usual mathematical method. In this case, it could be proven by contradiction. And since it is a direct theorem, its proof must be based not on the logical method of proof by contradiction, but on the usual mathematical method.

According to D. Abrarov, Academician V. I. Arnold, the most famous contemporary Russian mathematician, reacted to Wiles's proof "actively skeptical". The academician stated: “this is not real mathematics – real mathematics is geometric and has strong links with physics.”

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 does not prove Fermat's Last Theorem.

Fermat's Last Theorem is not proved with the help of the usual mathematical method, if it is given: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers, and if it is required 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 contributors, a specialist in number theory, made the following authoritative statement entitled: Brief retelling what Mirgorodsky did. I quote it verbatim:

« BUT. He proved that if z 2 \u003d x 2 + y , then 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 enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is performed (and only for it).

FROM. And then the author omitted the fact that from < in a subsequent degree may be = , not only > . A simple counterexample is the transition n=1 in n=2 in a Pythagorean triple.

D. This point does not contribute anything essential to the BTF proof. Conclusion: BTF has not been proven.”

I will consider his conclusion point by point.

BUT. In it, the BTF is proved for the entire infinite set of triples of Pythagorean numbers. Proven by a geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was opened, as I believe, by P. Fermat himself. Fermat might have had this in mind when he wrote:

"I have discovered a truly marvelous proof of this, but these margins are too narrow for it." This assumption of mine is based on the fact that in the Diophantine problem, against which, in the margins of the book, Fermat wrote, we are talking about solutions to the Diophantine equation, which are triples of Pythagorean numbers.

An infinite set of triples of Pythagorean numbers are solutions to the Diophatian 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 miraculous proof has a direct bearing on this fact. Later, Fermat could 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 proved nor disproved. It can be both accepted and rejected.

IN. In this section, I prove that, as a family of arbitrarily taken Pythagorean triple numbers, so the family of an arbitrarily taken non-Pythagorean triple of numbers BTF holds. This is a necessary, but insufficient and intermediate link in my proof of the BTF. The examples I have taken of the family of a triple of Pythagorean numbers and the family of a 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 enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is fulfilled (and only for it) is without foundation. He cannot refute the fact that I could just as well take other examples of Pythagorean and non-Pythagorean triples to get a specific family of one and the other triple.

Whatever pair of triples I take, checking their suitability for solving the problem can be carried out, in my opinion, only by the method of "simple enumeration". Any other method is not known to me and is not required. If he did not like Trotil, then he should have suggested another method, which he does not. Without offering anything in return, it is incorrect to condemn “simple enumeration”, which in this case is irreplaceable.

FROM. I omitted = between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 \u003d x 2 + y (1), in which the degree n > 2 — whole positive number. From the equality between the inequalities it follows obligatory consideration of equation (1) with a non-integer value of the degree n > 2 . Trotil counting compulsory consideration of equality between inequalities, actually considers necessary in the BTF proof, consideration of equation (1) with non-integer degree value n > 2 . I did this for myself and found that equation (1) with non-integer degree value n > 2 has a solution of three numbers: z, (z-1), (z-1) with a non-integer exponent.

It is unlikely that at least one year in the life of our editorial office passed without it receiving a good dozen proofs of Fermat's theorem. Now, after the “victory” over it, the flow has subsided, but has not dried up.

Of course, not to dry it completely, we publish this article. And not in my own defense - that, they say, that's why we kept silent, we ourselves have not matured yet to discuss such complex problems.

But if the article really seems complicated, look at the end of it right away. You will have to feel that the passions have calmed down temporarily, the science is not over, and soon new proofs of new theorems will be sent to the editors.

It seems that the 20th century was not in vain. First, people created a second Sun for a moment by detonating a hydrogen bomb. Then they walked on the moon and finally proved the notorious Fermat's theorem. Of these three miracles, the first two are on everyone's lips, for they have had enormous social consequences. On the contrary, the third miracle looks like another scientific toy - on a par with the theory of relativity, quantum mechanics and Gödel's theorem on the incompleteness of arithmetic. However, relativity and quanta led physicists to the hydrogen bomb, and the research of mathematicians filled our world with computers. Will this string of miracles continue into the 21st century? Is it possible to trace the connection between the next scientific toys and revolutions in our everyday life? Does this connection allow us to make successful predictions? Let's try to understand this using the example of Fermat's theorem.

Let's note for a start that she was born much later than her natural term. After all, the first special case of Fermat's theorem is the Pythagorean equation X 2 + Y 2 = Z 2 , relating the lengths of the sides of a right triangle. Having proved this formula twenty-five centuries ago, Pythagoras immediately asked himself the question: are there many triangles in nature in which both legs and hypotenuse have an integer length? It seems that the Egyptians knew only one such triangle - with sides (3, 4, 5). But it is not difficult to find other options: for example (5, 12, 13) , (7, 24, 25) or (8, 15, 17) . In all these cases, the length of the hypotenuse has the form (A 2 + B 2), where A and B are coprime numbers of different parity. In this case, the lengths of the legs are equal to (A 2 - B 2) and 2AB.

Noticing these relationships, Pythagoras easily proved that any triple of numbers (X \u003d A 2 - B 2, Y \u003d 2AB, Z \u003d A 2 + B 2) is a solution to the equation X 2 + Y 2 \u003d Z 2 and sets a rectangle with mutually simple side lengths. It is also seen that the number of different triples of this sort is infinite. But do all solutions of the Pythagorean equation have this form? Pythagoras was unable to prove or disprove such a hypothesis and left this problem to posterity without drawing attention to it. Who wants to highlight their failures? It seems that after this the problem of integral right-angled triangles lay in oblivion for seven centuries - until a new mathematical genius named Diophantus appeared in Alexandria.

We know little about him, but it is clear that he was nothing like Pythagoras. He felt like a king in geometry and even beyond - whether in music, astronomy or politics. The first arithmetic connection between the lengths of the sides of a harmonious harp, the first model of the Universe from concentric spheres carrying planets and stars, with the Earth in the center, and finally, the first republic of scientists in the Italian city of Crotone - these are the personal achievements of Pythagoras. What could Diophantus oppose to such successes - a modest researcher of the great Museum, which has long ceased to be the pride of the city crowd?

Only one thing: a better understanding of the ancient world of numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet master the positional notation of large numbers, but he knew what negative numbers were and probably spent many hours thinking about why the product of two negative numbers is positive. The world of integers was first revealed to Diophantus as a special universe, different from the world of stars, segments or polyhedra. The main occupation of scientists in this world is solving equations, a true master finds all possible solutions and proves that there are no other solutions. This is what Diophantus did with the quadratic Pythagorean equation, and then he thought: does at least one solution have a similar cubic equation X 3 + Y 3 = Z 3 ?

Diophantus failed to find such a solution; his attempt to prove that there are no solutions was also unsuccessful. Therefore, drawing up the results of his work in the book "Arithmetic" (it was the world's first textbook on number theory), Diophantus analyzed the Pythagorean equation in detail, but did not hint at a word about possible generalizations of this equation. But he could: after all, it was Diophantus who first proposed the notation for the powers of integers! But alas: the concept of “task book” was alien to Hellenic science and pedagogy, and publishing lists of unsolved problems was considered an indecent occupation (only Socrates acted differently). If you can't solve the problem - shut up! Diophantus fell silent, and this silence dragged on for fourteen centuries - until the onset of the New Age, when interest in the process of human thinking was revived.

Who didn’t fantasize about anything at the turn of the 16th-17th centuries! The indefatigable calculator Kepler tried to guess the connection between the distances from the Sun to the planets. Pythagoras failed. Kepler's success came after he learned how to integrate polynomials and other simple functions. On the contrary, the dreamer Descartes did not like long calculations, but it was he who first presented all points of the plane or space as sets of numbers. This audacious model reduces any geometric problem about figures to some algebraic problem about equations - and vice versa. For example, integer solutions of the Pythagorean equation correspond to integer points on the surface of a cone. The surface corresponding to the cubic equation X 3 + Y 3 = Z 3 looks more complicated, its geometric properties did not suggest anything to Pierre Fermat, and he had to pave new paths through the wilds of integers.

In 1636, a book by Diophantus, just translated into Latin from a Greek original, fell into the hands of a young lawyer from Toulouse, accidentally surviving in some Byzantine archive and brought to Italy by one of the Roman fugitives at the time of the Turkish ruin. Reading an elegant discussion of the Pythagorean equation, Fermat thought: is it possible to find such a solution, which consists of three square numbers? There are no small numbers of this kind: it is easy to verify this by enumeration. What about big decisions? Without a computer, Fermat could not carry out a numerical experiment. But he noticed that for each "large" solution of the equation X 4 + Y 4 = Z 4, one can construct a smaller solution. So the sum of the fourth powers of two integers is never equal to the same power of the third number! What about the sum of two cubes?

Inspired by the success for degree 4, Fermat tried to modify the "method of descent" for degree 3 - and succeeded. It turned out that it was impossible to compose two small cubes from those single cubes into which a large cube with an integer length of an edge fell apart. The triumphant Fermat made a brief note in the margins of Diophantus's book and sent a letter to Paris with a detailed report of his discovery. But he did not receive an answer - although usually mathematicians from the capital reacted quickly to the next success of their lone colleague-rival in Toulouse. What's the matter here?

Quite simply: by the middle of the 17th century, arithmetic had gone out of fashion. The great successes of the Italian algebraists of the 16th century (when polynomial equations of degrees 3 and 4 were solved) did not become the beginning of a general scientific revolution, because they did not allow solving new bright problems in adjacent fields of science. Now, if Kepler could guess the orbits of the planets using pure arithmetic ... But alas, this required mathematical analysis. This means that it must be developed - up to the complete triumph of mathematical methods in natural science! But analysis grows out of geometry, while arithmetic remains a field of play for idle lawyers and other lovers of the eternal science of numbers and figures.

So, Fermat's arithmetic successes turned out to be untimely and remained unappreciated. He was not upset by this: for the fame of a mathematician, the facts of differential calculus, analytic geometry and probability theory were revealed to him for the first time. All these discoveries of Fermat immediately entered the golden fund of the new European science, while number theory faded into the background for another hundred years - until it was revived by Euler.

This "king of mathematicians" of the 18th century was a champion in all applications of analysis, but he did not neglect arithmetic either, since new methods of analysis led to unexpected facts about numbers. Who would have thought that the infinite sum of inverse squares (1 + 1/4 + 1/9 + 1/16+…) is equal to π 2 /6? Who among the Hellenes could have foreseen that similar series would make it possible to prove the irrationality of the number π?

Such successes forced Euler to carefully reread the surviving manuscripts of Fermat (fortunately, the son of the great Frenchman managed to publish them). True, the proof of the “big theorem” for degree 3 has not been preserved, but Euler easily restored it just by pointing to the “descent method”, and immediately tried to transfer this method to the next prime degree - 5.

It wasn't there! In Euler's reasoning, complex numbers appeared that Fermat managed not to notice (such is the usual lot of discoverers). But the factorization of complex integers is a delicate matter. Even Euler did not fully understand it and put the "Fermat problem" aside, in a hurry to complete his main work - the textbook "Principles of Analysis", which was supposed to help every talented young man to stand on a par with Leibniz and Euler. The publication of the textbook was completed in St. Petersburg in 1770. But Euler did not return to Fermat's theorem, being sure that everything that his hands and mind touched would not be forgotten by the new scientific youth.

And so it happened: the Frenchman Adrien Legendre became Euler's successor in number theory. At the end of the 18th century, he completed the proof of Fermat's theorem for degree 5 - and although he failed for large prime powers, he compiled another textbook on number theory. May its young readers surpass the author in the same way that the readers of the Mathematical Principles of Natural Philosophy surpassed the great Newton! Legendre was no match for Newton or Euler, but there were two geniuses among his readers: Carl Gauss and Evariste Galois.

Such a high concentration of geniuses was facilitated by the French Revolution, which proclaimed the state cult of Reason. After that, every talented scientist felt like Columbus or Alexander the Great, able to discover or conquer a new world. Many succeeded, that is why in the 19th century scientific and technological progress became the main driver of the evolution of mankind, and all reasonable rulers (starting with Napoleon) were aware of this.

Gauss was close in character to Columbus. But he (like Newton) did not know how to captivate the imagination of rulers or students with beautiful speeches, and therefore limited his ambitions to the sphere of scientific concepts. Here he could do whatever he wanted. For example, the ancient problem of the trisection of an angle for some reason cannot be solved with a compass and straightedge. With the help of complex numbers depicting points of the plane, Gauss translates this problem into the language of algebra - and obtains a general theory of the feasibility of certain geometric constructions. Thus, at the same time, a rigorous proof of the impossibility of constructing a regular 7- or 9-gon with a compass and a ruler appeared, and such a way of constructing a regular 17-gon, which the wisest geometers of Hellas did not dream of.

Of course, such success is not given in vain: one has to invent new concepts that reflect the essence of the matter. Newton introduced three such concepts: flux (derivative), fluent (integral) and power series. They were enough to create mathematical analysis and the first scientific model of the physical world, including mechanics and astronomy. Gauss also introduced three new concepts: vector space, field, and ring. A new algebra grew out of them, subordinating Greek arithmetic and the theory of numerical functions created by Newton. It remained to subordinate the logic created by Aristotle to algebra: then it would be possible to prove the deducibility or non-derivability of any scientific statements from this set of axioms with the help of calculations! For example, does Fermat's theorem derive from the axioms of arithmetic, or does Euclid's postulate of parallel lines derive from other axioms of planimetry?

Gauss did not have time to realize this daring dream - although he advanced far and guessed the possibility of the existence of exotic (non-commutative) algebras. Only the daring Russian Nikolai Lobachevsky managed to build the first non-Euclidean geometry, and the first non-commutative algebra (Group Theory) was managed by the Frenchman Evariste Galois. And only much later than the death of Gauss - in 1872 - the young German Felix Klein guessed that the variety of possible geometries can be brought into one-to-one correspondence with the variety of possible algebras. Simply put, every geometry is defined by its symmetry group - while general algebra studies all possible groups and their properties.

But such an understanding of geometry and algebra came much later, and the assault on Fermat's theorem resumed during Gauss's lifetime. He himself neglected Fermat's theorem out of the principle: it is not the king's business to solve individual problems that do not fit into a bright scientific theory! But the students of Gauss, armed with his new algebra and the classical analysis of Newton and Euler, reasoned differently. First, Peter Dirichlet proved Fermat's theorem for degree 7 using the ring of complex integers generated by the roots of this degree of unity. Then Ernst Kummer extended the Dirichlet method to EVERYTHING simple powers(!) - so it seemed to him rashly, and he triumphed. But soon a sobering up came: the proof passes flawlessly only if every element of the ring is uniquely decomposed into prime factors! For ordinary integers, this fact was already known to Euclid, but only Gauss gave its rigorous proof. But what about the whole complex numbers?

According to the “principle of the greatest mischief”, there can and SHOULD occur an ambiguous factorization! As soon as Kummer learned to calculate the degree of ambiguity by methods of mathematical analysis, he discovered this dirty trick in the ring for degree 23. Gauss did not have time to learn about this version of exotic commutative algebra, but Gauss's students grew a new beautiful Theory of Ideals in place of another dirty trick. True, this did not help much in solving Fermat's problem: only its natural complexity became clearer.

Throughout the 19th century, this ancient idol demanded more and more sacrifices from its admirers in the form of new complex theories. It is not surprising that by the beginning of the 20th century, believers became discouraged and rebelled, rejecting their former idol. The word "fermatist" has become a pejorative term among professional mathematicians. And although for complete proof Fermat's theorem was awarded a considerable prize, but its applicants were mostly self-confident ignoramuses. The strongest mathematicians of that time - Poincaré and Hilbert - defiantly eschewed this topic.

In 1900, Hilbert did not include Fermat's Theorem in the list of twenty-three major problems facing the mathematics of the twentieth century. True, he included in their series the general problem of the solvability of Diophantine equations. The hint was clear: follow the example of Gauss and Galois, create general theories of new mathematical objects! Then one fine (but not predictable in advance) day, the old splinter will fall out by itself.

This is how the great romantic Henri Poincaré acted. Neglecting many "eternal" problems, all his life he studied the SYMMETRIES of certain objects of mathematics or physics: either functions of a complex variable, or trajectories of motion of celestial bodies, or algebraic curves or smooth manifolds (these are multidimensional generalizations of curved lines). The motive for his actions was simple: if two different objects have similar symmetries, it means that there is an internal relationship between them, which we are not yet able to comprehend! For example, each of the two-dimensional geometries (Euclid, Lobachevsky or Riemann) has its own symmetry group, which acts on the plane. But the points of the plane are complex numbers: in this way the action of any geometric group is transferred to the vast world of complex functions. It is possible and necessary to study the most symmetrical of these functions: AUTOMORPHOUS (which are subject to the Euclid group) and MODULAR (which are subject to the Lobachevsky group)!

There are also elliptic curves in the plane. They have nothing to do with the ellipse, but are given by equations of the form Y 2 = AX 3 + BX 2 + CX and therefore intersect with any straight line at three points. This fact allows us to introduce multiplication among the points of an elliptic curve - to turn it into a group. The algebraic structure of this group reflects the geometric properties of the curve; perhaps it is uniquely determined by its group? This question is worth studying, since for some curves the group of interest to us turns out to be modular, that is, it is related to the Lobachevsky geometry ...

This is how Poincaré reasoned, seducing the mathematical youth of Europe, but at the beginning of the 20th century these temptations did not lead to bright theorems or hypotheses. It turned out differently with Hilbert's call: to study the general solutions of Diophantine equations with integer coefficients! In 1922, the young American Lewis Mordell connected the set of solutions of such an equation (this is a vector space of a certain dimension) with the geometric genus of the complex curve that is given by this equation. Mordell came to the conclusion that if the degree of the equation is sufficiently large (more than two), then the dimension of the solution space is expressed in terms of the genus of the curve, and therefore this dimension is FINITE. On the contrary - to the power of 2, the Pythagorean equation has an INFINITE-DIMENSIONAL family of solutions!

Of course, Mordell saw the connection of his hypothesis with Fermat's theorem. If it becomes known that for every degree n > 2 the space of entire solutions of Fermat's equation is finite-dimensional, this will help to prove that there are no such solutions at all! But Mordell did not see any way to prove his hypothesis - and although he lived a long life, he did not wait for the transformation of this hypothesis into Faltings' theorem. This happened in 1983, in a completely different era, after the great successes of the algebraic topology of manifolds.

Poincaré created this science as if by accident: he wanted to know what three-dimensional manifolds are. After all, Riemann figured out the structure of all closed surfaces and got a very simple answer! If there is no such answer in a three-dimensional or multidimensional case, then you need to come up with a system of algebraic invariants of the manifold that determines its geometric structure. It is best if such invariants are elements of some groups - commutative or non-commutative.

Strange as it may seem, this audacious plan by Poincaré succeeded: it was carried out from 1950 to 1970 thanks to the efforts of a great many geometers and algebraists. Until 1950, there was a quiet accumulation of various methods for classifying manifolds, and after this date, a critical mass of people and ideas seemed to have accumulated and an explosion occurred, comparable to the invention of mathematical analysis in the 17th century. But the analytic revolution lasted for a century and a half, covering the creative biographies of four generations of mathematicians - from Newton and Leibniz to Fourier and Cauchy. On the contrary, the topological revolution of the 20th century was within twenty years, thanks to the large number of its participants. At the same time, a large generation of self-confident young mathematicians has emerged, suddenly left without work in their historical homeland.

In the seventies they rushed into the adjacent fields of mathematics and theoretical physics. Many have created their own scientific schools in dozens of universities in Europe and America. Many students of different ages and nationalities, with different abilities and inclinations, still circulate between these centers, and everyone wants to be famous for some discovery. It was in this pandemonium that Mordell's conjecture and Fermat's theorem were finally proven.

However, the first swallow, unaware of its fate, grew up in Japan in the hungry and unemployed post-war years. The name of the swallow was Yutaka Taniyama. In 1955, this hero turned 28 years old, and he decided (together with friends Goro Shimura and Takauji Tamagawa) to revive mathematical research in Japan. Where to begin? Of course, with overcoming isolation from foreign colleagues! So in 1955, three young Japanese hosted the first international conference on algebra and number theory in Tokyo. It was apparently easier to do this in Japan reeducated by the Americans than in Russia frozen by Stalin ...

Among the guests of honor were two heroes from France: Andre Weil and Jean-Pierre Serre. Here the Japanese were very lucky: Weyl was the recognized head of the French algebraists and a member of the Bourbaki group, and the young Serre played a similar role among topologists. In heated discussions with them, the heads of the Japanese youth cracked, their brains melted, but in the end, such ideas and plans crystallized that could hardly have been born in a different environment.

One day, Taniyama approached Weil with a question about elliptic curves and modular functions. At first, the Frenchman did not understand anything: Taniyama was not a master of speaking English. Then the essence of the matter became clear, but Taniyama did not manage to give his hopes an exact formulation. All Weil could reply to the young Japanese was that if he were very lucky in terms of inspiration, then something sensible would grow out of his vague hypotheses. But while the hope for it is weak!

Obviously, Weil did not notice the heavenly fire in Taniyama's gaze. And there was fire: it seems that for a moment the indomitable thought of the late Poincaré moved into the Japanese! Taniyama came to believe that every elliptic curve is generated by modular functions - more precisely, it is "uniformized by a modular form". Alas, this exact wording was born much later - in Taniyama's conversations with his friend Shimura. And then Taniyama committed suicide in a fit of depression... His hypothesis was left without an owner: it was not clear how to prove it or where to test it, and therefore no one took it seriously for a long time. The first response came only thirty years later - almost like in Fermat's era!

The ice broke in 1983, when twenty-seven-year-old German Gerd Faltings announced to the whole world: Mordell's conjecture had been proven! Mathematicians were on their guard, but Faltings was a true German: there were no gaps in his long and complicated proof. It's just that the time has come, the facts and concepts have accumulated - and now one talented algebraist, relying on the results of ten other algebraists, has managed to solve a problem that has stood waiting for the master for sixty years. This is not uncommon in 20th-century mathematics. It is worth recalling the secular continuum problem in set theory, Burnside's two conjectures in group theory, or the Poincaré conjecture in topology. Finally, in number theory, the time has come to harvest the old crops ... Which top will be the next in a series of conquered mathematicians? Will Euler's problem, Riemann's hypothesis, or Fermat's theorem collapse? It would be good!

And now, two years after the revelation of Faltings, another inspired mathematician appeared in Germany. His name was Gerhard Frey, and he claimed something strange: that Fermat's theorem is DERIVED from Taniyama's conjecture! Unfortunately, Frey's style of expressing his thoughts was more reminiscent of the unfortunate Taniyama than his clear compatriot Faltings. In Germany, no one understood Frey, and he went overseas - to the glorious town of Princeton, where, after Einstein, they got used to not such visitors. No wonder Barry Mazur, a versatile topologist, one of the heroes of the recent assault on smooth manifolds, made his nest there. And a student grew up next to Mazur - Ken Ribet, equally experienced in the intricacies of topology and algebra, but still not glorifying himself in any way.

When he first heard Frey's speeches, Ribet decided that this was nonsense and near-science fiction (probably, Weil reacted to Taniyama's revelations in the same way). But Ribet could not forget this "fantasy" and at times returned to it mentally. Six months later, Ribet believed that there was something sensible in Frey's fantasies, and a year later he decided that he himself could almost prove Frey's strange hypothesis. But some "holes" remained, and Ribet decided to confess to his boss Mazur. He listened attentively to the student and calmly replied: “Yes, you have done everything! Here you need to apply the transformation Ф, here - use Lemmas B and K, and everything will take on an impeccable form! So Ribet made a leap from obscurity to immortality, using a catapult in the person of Frey and Mazur. In fairness, all of them - along with the late Taniyama - should be considered proofs of Fermat's Last Theorem.

But here's the problem: they derived their statement from the Taniyama hypothesis, which itself has not been proven! What if she's unfaithful? Mathematicians have long known that “anything follows from a lie”, if Taniyama’s guess is wrong, then Ribet’s impeccable reasoning is worthless! We urgently need to prove (or disprove) Taniyama's conjecture - otherwise someone like Faltings will prove Fermat's theorem in a different way. He will become a hero!

It is unlikely that we will ever know how many young or seasoned algebraists jumped on Fermat's theorem after the success of Faltings or after the victory of Ribet in 1986. All of them tried to work in secret, so that in case of failure they would not be ranked among the community of “dummies”-fermatists. It is known that the most successful of all - Andrew Wiles from Cambridge - felt the taste of victory only at the beginning of 1993. This not so much pleased as frightened Wiles: what if his proof of the Taniyama conjecture showed an error or a gap? Then his scientific reputation perished! You have to carefully write down the proof (but it will be many dozens of pages!) And put it aside for six months or a year, so that later you can re-read it cold-bloodedly and meticulously ... But what if someone publishes their proof during this time? Oh trouble...

Yet Wiles came up with a double way to quickly test his proof. First, you need to trust one of your reliable friends and colleagues and tell him the whole course of reasoning. From the outside, all the mistakes are more visible! Secondly, it is necessary to read a special course on this topic to smart students and graduate students: these smart people will not miss a single lecturer's mistake! Just do not tell them the ultimate goal of the course until the last moment - otherwise the whole world will know about it! And of course, you need to look for such an audience away from Cambridge - it’s better not even in England, but in America ... What could be better than distant Princeton?

Wiles went there in the spring of 1993. His patient friend Niklas Katz, after listening to Wiles' long report, found a number of gaps in it, but all of them were easily corrected. But the Princeton graduate students soon ran away from Wiles's special course, not wanting to follow the whimsical thought of the lecturer, who leads them to no one knows where. After such a (not particularly deep) review of his work, Wiles decided that it was time to reveal a great miracle to the world.

In June 1993, another conference was held in Cambridge, dedicated to the "Iwasawa theory" - a popular section of number theory. Wiles decided to tell his proof of the Taniyama conjecture on it, without announcing the main result until the very end. The report went on for a long time, but successfully, journalists gradually began to flock, who sensed something. Finally, thunder struck: Fermat's theorem is proved! The general rejoicing was not overshadowed by any doubts: everything seems to be clean ... But two months later, Katz, having read the final text of Wiles, noticed another gap in it. A certain transition in reasoning relied on the "Euler system" - but what Wiles built was not such a system!

Wiles checked the bottleneck and realized that he was mistaken here. Even worse: it is not clear how to replace the erroneous reasoning! This was followed by the darkest months of Wiles' life. Previously, he freely synthesized an unprecedented proof from the material at hand. Now he is tied to a narrow and clear task - without the certainty that it has a solution and that he will be able to find it in the foreseeable future. Recently, Frey could not resist the same struggle - and now his name was obscured by the name of the lucky Ribet, although Frey's guess turned out to be correct. And what will happen to MY guess and MY name?

This hard labor lasted exactly one year. In September 1994, Wiles was ready to admit defeat and leave the Taniyama hypothesis to more fortunate successors. Having made such a decision, he began to slowly reread his proof - from beginning to end, listening to the rhythm of reasoning, re-experiencing the pleasure of successful discoveries. Having reached the "damned" place, Wiles, however, did not mentally hear a false note. Was the course of his reasoning still impeccable, and the error arose only in the VERBAL description of the mental image? If there is no “Euler system” here, then what is hidden here?

Suddenly, a simple thought came to me: the "Euler system" does not work where the Iwasawa theory is applicable. Why not apply this theory directly - fortunately, it is close and familiar to Wiles himself? And why did he not try this approach from the very beginning, but got carried away by someone else's vision of the problem? Wiles could no longer remember these details - and it became useless. He carried out the necessary reasoning within the framework of the Iwasawa theory, and everything turned out in half an hour! Thus - with a delay of one year - the last gap in the proof of Taniyama's conjecture was closed. The final text was given to the mercy of a group of reviewers of the most famous mathematical journal, a year later they declared that now there are no errors. Thus, in 1995, Fermat's last conjecture died at the age of 360, turning into a proven theorem, which will inevitably enter the number theory textbooks.

Summing up the three-century fuss around Fermat's theorem, we have to draw a strange conclusion: this heroic epic could not have happened! Indeed, the Pythagorean theorem expresses a simple and important connection between visual natural objects - the lengths of segments. But the same cannot be said of Fermat's Theorem. It looks more like a cultural superstructure on a scientific substrate - like reaching the North Pole of the Earth or flying to the moon. Let us recall that both of these feats were sung by writers long before they were accomplished - back in ancient times, after the appearance of Euclid's "Elements", but before the appearance of Diophantus's "Arithmetic". So, then there was a public need for intellectual exploits of this kind - at least imaginary! Previously, the Hellenes had had enough of Homer's poems, just as a hundred years before Fermat, the French had had enough of religious passions. But then religious passions subsided - and science stood next to them.

In Russia, such processes began a hundred and fifty years ago, when Turgenev put Yevgeny Bazarov on a par with Yevgeny Onegin. True, the writer Turgenev poorly understood the motives for the actions of the scientist Bazarov and did not dare to sing them, but this was soon done by the scientist Ivan Sechenov and the enlightened journalist Jules Verne. The spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and here comes science fiction first, and then popular science literature (including the magazine "Knowledge is Power").

At the same time, a specific scientific topic is not at all important for the general public and is not very important even for the heroes-performers. So, having heard about the achievement of the North Pole by Peary and Cook, Amundsen instantly changed the goal of his already prepared expedition - and soon reached the South Pole, ahead of Scott by one month. Later, Yuri Gagarin's successful circumnavigation of the Earth forced President Kennedy to change the former goal of the American space program to a more expensive but far more impressive one: landing men on the moon.

Even earlier, the insightful Hilbert answered the naive question of students: “The solution of what scientific problem would be most useful now”? - answered with a joke: “Catch a fly on the far side of the moon!” To the perplexed question: “Why is this necessary?” - followed by a clear answer: “Nobody needs THIS! But think of the scientific methods and technical means that we will have to develop to solve such a problem - and what a lot of other beautiful problems we will solve along the way!

This is exactly what happened with Fermat's Theorem. Euler could well have overlooked it.

In this case, some other problem would become the idol of mathematicians - perhaps also from number theory. For example, the problem of Eratosthenes: is there a finite or infinite set of twin primes (such as 11 and 13, 17 and 19, and so on)? Or Euler's problem: is every even number the sum of two prime numbers? Or: is there an algebraic relation between the numbers π and e? These three problems have not yet been solved, although in the 20th century mathematicians have come close to understanding their essence. But this century also gave rise to many new, no less interesting problems, especially at the intersection of mathematics with physics and other branches of natural science.

Back in 1900, Hilbert singled out one of them: to create a complete system of axioms of mathematical physics! A hundred years later, this problem is far from being solved, if only because the arsenal of mathematical means of physics is steadily growing, and not all of them have a rigorous justification. But after 1970, theoretical physics split into two branches. One (classical) since the time of Newton has been modeling and predicting STABLE processes, the other (newborn) is trying to formalize the interaction of UNSTABLE processes and ways to control them. It is clear that these two branches of physics must be axiomatized separately.

The first of them will probably be dealt with in twenty or fifty years ...

And what is missing from the second branch of physics - the one that is in charge of all kinds of evolution (including outlandish fractals and strange attractors, the ecology of biocenoses and Gumilyov's theory of passionarity)? This we are unlikely to understand soon. But the worship of scientists to the new idol has already become a mass phenomenon. Probably, an epic will unfold here, comparable to the three-century biography of Fermat's theorem. Thus, at the intersection of different sciences, new idols are born - similar to religious ones, but more complex and dynamic ...

Apparently, a person cannot remain a person without overthrowing the old idols from time to time and without creating new ones - in pain and with joy! Pierre Fermat was lucky to be at a fateful moment close to the hot spot of the birth of a new idol - and he managed to leave an imprint of his personality on the newborn. One can envy such a fate, and it is not a sin to imitate it.

Sergei Smirnov
"Knowledge is power"

Grigory Perelman. Refusenik

Vasily Maksimov

In August 2006, the names of the best mathematicians on the planet were announced, who received the most prestigious Fields Medal - a kind of analogue of the Nobel Prize, which mathematicians, at the whim of Alfred Nobel, were deprived of. The Fields Medal - in addition to the badge of honor, laureates are awarded a check for fifteen thousand Canadian dollars - is awarded by the International Congress of Mathematicians every four years. It was established by Canadian scientist John Charles Fields and was first awarded in 1936. Since 1950, the Fields Medal has been awarded regularly personally by the King of Spain for his contribution to the development of mathematical science. From one to four scientists under the age of forty can become laureates of the award. Forty-four mathematicians have already received the prize, including eight Russians.

Grigory Perelman. Henri Poincare.

In 2006, the Frenchman Wendelin Werner, the Australian Terence Tao and two Russians, Andrey Okounkov, who works in the USA, and Grigory Perelman, a scientist from St. Petersburg, became laureates. However, at the last moment it became known that Perelman refused this prestigious award - as the organizers announced, "for reasons of principle."

Such an extravagant act of the Russian mathematician did not come as a surprise to people who knew him. This is not the first time he refuses mathematical awards, explaining his decision by the fact that he does not like solemn events and excessive hype around his name. Ten years ago, in 1996, Perelman refused the prize of the European Mathematical Congress, citing the fact that he had not finished work on the scientific problem nominated for the award, and this was not the last case. The Russian mathematician seems to have made it his life's goal to surprise people, going against public opinion and the scientific community.

Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad. From a young age he was fond of the exact sciences, graduated with brilliance from the famous 239th secondary school with in-depth study of mathematics, won numerous mathematical olympiads: for example, in 1982, as part of a team of Soviet schoolchildren, he participated in the International Mathematical Olympiad, held in Budapest. Perelman without exams was enrolled in the mechanics and mathematics department of Leningrad University, where he studied "excellently", continuing to win in mathematical competitions at all levels. After graduating from the university with honors, he entered graduate school at the St. Petersburg Department of the Steklov Mathematical Institute. His supervisor was the famous mathematician Academician Aleksandrov. Having defended his Ph.D. thesis, Grigory Perelman remained at the institute, in the laboratory of geometry and topology. Known for his work on the theory of Alexandrov spaces, he was able to find evidence for a number of important hypotheses. Despite numerous offers from leading Western universities, Perelman prefers to work in Russia.

His most notorious success was the solution in 2002 of the famous Poincare conjecture, published in 1904 and since then remained unproven. Perelman worked on it for eight years. The Poincaré hypothesis was considered one of the greatest mathematical mysteries, and its solution was considered the most important achievement in mathematical science: it will instantly advance the study of the problems of the physical and mathematical foundations of the universe. The brightest minds on the planet predicted its solution only in a few decades, and the Clay Institute of Mathematics in Cambridge, Massachusetts, made the Poincaré problem one of the seven most interesting unsolved mathematical problems of the millennium, each of which was promised a million dollar prize (Millennium Prize Problems) .

The hypothesis (sometimes called the problem) of the French mathematician Henri Poincaré (1854–1912) is formulated as follows: any closed, simply connected three-dimensional space is homeomorphic to a three-dimensional sphere. For clarification, a good example is used: if you wrap an apple with a rubber band, then, in principle, by pulling the tape together, you can squeeze the apple into a point. If you wrap a donut with the same tape, then you cannot squeeze it into a point without tearing either the donut or rubber. In this context, an apple is called a "singly connected" figure, but a donut is not simply connected. Almost a hundred years ago, Poincaré established that the two-dimensional sphere is simply connected and suggested that the three-dimensional sphere is also simply connected. The best mathematicians in the world could not prove this conjecture.

To qualify for the Clay Institute prize, Perelman only needed to publish his solution in one of the scientific journals, and if within two years no one can find an error in his calculations, then the solution will be considered correct. However, Perelman deviated from the rules from the very beginning, publishing his solution on the preprint site of the Los Alamos Science Laboratory. Perhaps he was afraid that an error had crept into his calculations - a similar story had already happened in mathematics. In 1994, the English mathematician Andrew Wiles proposed a solution to the famous Fermat's theorem, and a few months later it turned out that an error had crept into his calculations (although it was later corrected, and the sensation still took place). There is still no official publication of the proof of the Poincare conjecture - but there is an authoritative opinion of the best mathematicians on the planet, confirming the correctness of Perelman's calculations.

The Fields Medal was awarded to Grigory Perelman precisely for solving the Poincaré problem. But the Russian scientist refused the prize, which he undoubtedly deserves. “Grigory told me that he feels isolated from the international mathematical community, outside this community, therefore he does not want to receive an award,” John Ball, the president of the World Union of Mathematicians (WCM), said at a press conference in Madrid.

There are rumors that Grigory Perelman is going to leave science altogether: six months ago he quit his native Steklov Mathematical Institute, and they say that he will no longer do mathematics. Perhaps the Russian scientist believes that by proving the famous hypothesis, he has done everything he could for science. But who will undertake to talk about the train of thought of such a bright scientist and extraordinary person? .. Perelman refuses any comments, and he told The Daily Telegraph newspaper: “Nothing that I can say is of the slightest public interest.” However, the leading scientific publications were unanimous in their assessments when they reported that "Grigory Perelman, having solved the Poincare theorem, stood on a par with the greatest geniuses of the past and present."

Monthly literary and journalistic magazine and publishing house.

File FERMA-KDVar © N. M. Koziy, 2008

Certificate of Ukraine No. 27312

A BRIEF PROOF OF FERMAT'S GREAT THEOREM

Fermat's Last Theorem is formulated as follows: Diophantine equation (http://soluvel.okis.ru/evrika.html):

BUT n + V n = C n * /1/

where n- a positive integer greater than two has no solution in positive integers A , B , FROM .

PROOF

From the formulation of Fermat's Last Theorem it follows: if n is a positive integer greater than two, then, provided that two of the three numbers BUT , IN or FROM are positive integers, one of these numbers is not a positive integer.

We build the proof on the basis of the fundamental theorem of arithmetic, which is called the "theorem on the uniqueness of factorization" or "theorem on the uniqueness of factorization of integer composite numbers". Odd and even exponents possible n . Let's consider both cases.

1. Case One: Exponent n - odd number.

In this case, the expression /1/ is converted according to known formulas as follows:

BUT n + IN n = FROM n /2/

We believe that A And B are positive integers.

Numbers BUT , IN And FROM must be relatively prime numbers.

From equation /2/ it follows that for given values ​​of numbers A And B factor ( A + B ) n , FROM.

Let's say the number FROM - a positive integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition :

FROM n = A n + B n =(A+B) n ∙ D n , / 3/

where is the multiplier D n D

From equation /3/ it follows:

Equation /3/ also implies that the number [ C n = A n + B n ] provided that the number FROM ( A + B ) n. However, it is known that:

A n + B n < ( A + B ) n /5/

Consequently:

is a fractional number less than one. /6/

A fractional number.

n

For odd exponents n >2 number:

< 1- дробное число, не являющееся рациональной дробью.

From the analysis of the equation /2/ it follows that with an odd exponent n number:

FROM n = BUT n + IN n = (A+B)

consists of two definite algebraic factors, and for any value of the exponent n the algebraic factor remains unchanged ( A + B ).

Thus, Fermat's Last Theorem has no solution in positive integers for an odd exponent n >2.

2. Case Two: Exponent n - even number .

The essence of Fermat's last theorem will not change if the equation /1/ is rewritten as follows:

A n = C n - B n /7/

In this case, the equation /7/ is transformed as follows:

A n = C n - B n = ( FROM +B)∙(C n-1 + C n-2 B+ C n-3 ∙ B 2 +…+ C ∙ B n -2 + B n -1 ). /8/

We accept that FROM And IN- whole numbers.

From equation /8/ it follows that for given values ​​of numbers B And C factor (C+ B ) has the same value for any value of the exponent n , hence it is a divisor of a number A .

Let's say the number BUT is an integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition :

BUT n = C n - B n =(C+ B ) n ∙ D n , / 9/

where is the multiplier D n must be an integer and therefore a number D must also be an integer.

From equation /9/ it follows:

/10/

Equation /9/ also implies that the number [ BUT n = FROM n - B n ] provided that the number BUT- an integer, must be divisible by a number (C+ B ) n. However, it is known that:

FROM n - B n < (С+ B ) n /11/

Consequently:

is a fractional number less than one. /12/

A fractional number.

It follows that for an odd value of the exponent n equation /1/ of Fermat's last theorem has no solution in positive integers.

With even exponents n >2 number:

< 1- дробное число, не являющееся рациональной дробью.

Thus, Fermat's Last Theorem has no solution in positive integers and for an even exponent n >2.

The general conclusion follows from the above: the equation /1/ of Fermat's last theorem has no solution in positive integers A, B And FROM provided that the exponent n>2.

ADDITIONAL REASONS

In the case when the exponent n – even number, algebraic expression ( C n - B n ) decomposed into algebraic factors:

C 2 - B 2 \u003d(C-B) ∙ (C+B); /13/

C 4 – B 4 = ( C-B) ∙ (C+B) (C 2 + B 2);/14/

C 6 - B 6 =(C-B) ∙ (C + B) (C 2 -CB + B 2) ∙ (C 2 + CB + B 2) ; /15/

C 8 - B 8= (C-B) ∙ (C+B) ∙ (C 2 + B 2) ∙ (C 4 + B 4)./16/

Let's give examples in numbers.

EXAMPLE 1: B=11; C=35.

C 2 – B 2 = (2 2 ∙ 3) ∙ (2 23) = 2 4 3 23;

C 4 – B 4 = (2 2 ∙ 3) ∙ (2 23) (2 673) = 2 4 3 23 673;

C 6 – B 6 = (2 2 ∙ 3) ∙ (2 23) (31 2) (3 577) =2 ∙ 3 ​​∙ 23 ∙ 31 2 ∙ 577;

C 8 – B 8 = (2 2 ∙ 3) ∙ (2 23) (2 673) ∙ (2 75633) = 2 5 ∙ 3 ∙ 23 ∙673 ∙ 75633 .

EXAMPLE 2: B=16; C=25.

C 2 – B 2 = (3 2) ∙ (41) = 3 2 ∙ 41;

C 4 – B 4 = (3 2) ∙ (41) (881) =3 2 ∙ 41 881;

C 6 – B 6 = (3 2) ∙ (41) ∙ (2 2 ∙ 3) ∙ (13 37) (3 ∙ 7 61) = 3 3 7 ∙ 13 37 ∙ 41 ∙ 61;

C 8 – B 8 = (3 2) ∙ (41) ∙ (881) ∙ (17 26833) = 3 2 ∙ 41 ∙ 881 ∙ 17 26833.

From the analysis of equations /13/, /14/, /15/ and /16/ and their corresponding numerical examples, it follows:

For a given exponent n , if it's an even number, a number BUT n = C n - B n decomposes into a well-defined number of well-defined algebraic factors;

For any degree n , if it is an even number, in algebraic expression ( C n - B n ) there are always multipliers ( C - B ) And ( C + B ) ;

Each algebraic factor corresponds to a well-defined numerical factor;

For given values ​​of numbers IN And FROM numeric factors can be prime numbers or composite numeric factors;

Each composite numerical factor is a product of prime numbers, which are partially or completely absent from other composite numerical factors;

The value of prime numbers in the composition of composite numerical factors increases with the increase in these factors;

The composition of the largest composite numerical factor corresponding to the largest algebraic factor includes the largest prime number in a power less than the exponent n(most often in the first degree).

CONCLUSIONS: additional justifications support the conclusion that Fermat's Last Theorem has no solution in positive integers.

mechanical engineer