Pierre Fermat, reading "Arithmetic" by Diophantus of Alexandria and reflecting on its tasks, had the habit of writing down the results of his reflections in the margins of the book in the form of short remarks. 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 biquadrat into two biquadrats, and, in general, no degree greater than a square by two degrees with the same exponent. I have discovered a truly wonderful proof of this, but these fields are too narrow for him.» / 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 Diophantus problem with the modern formulation of Fermat's great theorem, which has the form of an equation.
« The equation
x n + y n \u003d z n (where n is an integer greater than two)
has no solution in positive integers»
The commentary 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 commentary can be interpreted as follows: if a quadratic equation with three unknowns has an infinite set of solutions on the set of all triples of Pythagorean numbers, then, on the contrary, an equation with three unknowns to a degree greater than the square
There is not even a hint of its connection with the problem of Diophantus in the equation. Its assertion requires proof, but under it there is no 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.
- The equation of Fermat's theorem is taken as its conclusion, the validity of which is verified with the help of the proof.
- The same equation is called original the equation from which its proof must proceed.
As a result, a tautology was formed: “ If the equation has no solutions in positive integers, then it has no solutions in positive integers"The proof of the tautology is deliberately incorrect and devoid of any sense. But it is proved by contradiction.
- The opposite assumption is made to that of the equation you want to prove. It should not contradict the original equation, but it contradicts it. It makes no sense to prove what is accepted without proof, and to accept without proof what is required to be proved.
- 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 unrealizable dream of specialists and amateurs 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 \u003d 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 \u003d z n
where n\u003e 2
(2) has no solutions on the set of positive integers. "
Evidence.
AND) Everyone knows that equation (1) has an infinite set of solutions on the set of all triples of Pythagorean numbers. Let us prove that no 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, the sides of equation (1) are interchanged. Pythagorean numbers (z, x, y) can be interpreted as the lengths of the sides of a right-angled triangle, and the squares ( x 2, y 2, z 2) can be interpreted as the area of \u200b\u200bsquares built on its hypotenuse and legs.
The squares of the squares of equation (1) are multiplied by an arbitrary height h :
z 2 h \u003d 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 \u003d z :
z 3 \u003d x 2 z + y 2 z (4)
The volume of the cube is decomposed into two volumes of two parallelepipeds. 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\u003e x 3 + y 3 (5)
On the set of triples of Pythagorean numbers ( x, y, z ) at n \u003d 3 there can be no solution to equation (2). Therefore, 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 \u003d z 2 :
z 2 z 2 \u003d 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.
Leave the left side of equation (6) unchanged. On its right side is the height z 2
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.
Leave the left side of equation (8) unchanged.
On the right side the height z n-2
reduce to x n-2
in the first term and decrease to y n-2
in the second term. Equation (8) turns into the inequality:
| z n\u003e x n + y n | (9) |
On the set of triples of Pythagorean numbers, there can be no solution to equation (2).
Consequently, on the set of all triples of Pythagorean numbers for all n\u003e 2 equation (2) has no solutions.
Received "postinno miraculous proof", 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 Eq. (2) has no solutions on the set of triples of non-Pythagorean numbers, which is a failure of the family of an arbitrary triple of Pythagorean numbers z \u003d 13, x \u003d 12, y \u003d 5 and the family of an arbitrary triple of positive integers z \u003d 21, x \u003d 19, y \u003d 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, that is, 78 and 210.
Each member of family (10) contains z \u003d 13 and variables x and at 13\u003e x\u003e 0 , 13\u003e y\u003e 0 1
Each member of family (11) contains z \u003d 21 and variables x and at that take integer values 21\u003e x\u003e 0 , 21\u003e y\u003e 0 ... The variables gradually decrease by 1 .
The triples of numbers in the sequence (10) and (11) can be represented as a sequence of third degree inequalities:
| 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 fourth degree inequalities:
| 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 confirmed by the elevation of 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 more than the sum of the cubes of the two lesser numbers.
The biquadrat of a larger number cannot be decomposed into two biquadrats of smaller numbers. It is either less or more than the sum of the biquadrats of smaller numbers.
As the exponent increases, all inequalities, except for the left extreme inequality, have the same meaning:
Inequalities, they all have the same meaning: the degree of a larger number is greater than the sum of the powers of less than two numbers with the same exponent:
| 13 n\u003e 12 n + 12 n; 13 n\u003e 12 n + 11 n; ...; 13 n\u003e 7 n + 4 n; ...; 13 n\u003e 1 n + 1 n | (12) | |
| 21 n\u003e 20 n + 20 n; 21 n\u003e 20 n + 19 n; ...; ;…; 21 n\u003e 1 n + 1 n | (13) |
The leftmost term of sequences (12) (13) is the weakest inequality. Its correctness determines the correctness of all subsequent inequalities of sequence (12) for n\u003e 8 and sequence (13) for n\u003e 14 .
There cannot be a single equality among them. An arbitrary triple of positive integers (21,19,16) is not a solution to equation (2) of Fermat's great theorem. If an arbitrarily taken 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 we had to prove.
FROM) Fermat's commentary on the Diophantus problem states that it is impossible to decompose “ in general, no degree greater than the square, by two degrees with the same exponent».
Whole a degree greater than a square is really impossible to decompose into two degrees with the same exponent. Inappropriate a degree greater than a square can be decomposed into two degrees with the same exponent.
Any arbitrary triple of positive integers (z, x, y) can 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 obtained inequalities can be represented as 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 in which the left side is less than the right side and ends with inequalities in which the right side is less than the left side. With increasing exponent n\u003e 2 the number of inequalities on the right-hand side of sequence (14) increases. With an exponent n \u003d k all the 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 in the sequence (14). As a result of an increase in the exponent for all inequalities, the left side turns out to be larger than the right side:
| z k\u003e (z-1) k + (z-1) k; z k\u003e (z-1) k + (z-2) k; ...; z k\u003e 2 k + 1 k; z k\u003e 1 k + 1 k | (15) |
With a further increase in the exponent n\u003e 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\u003e 2 , z\u003e x , z\u003e y
In an arbitrary triple of positive integers z can be an arbitrarily large natural number. For all natural numbers that are not greater than z , Fermat's Last Theorem is proved.
D) No matter how large the number z , in the natural series of numbers before it there is a large, but finite set of integers, and after it - an infinite set of integers.
Let us prove that the whole infinite set of natural numbers greater than z , form triples of numbers that are not solutions to the equation of Great Fermat's Theorem, for example, an arbitrary triple of positive integers (z + 1, x, y) , wherein z + 1\u003e x and z + 1\u003e y for all values \u200b\u200bof the exponent n\u003e 2 is not a solution to the equation of the Great Fermat's theorem.
An arbitrary triple of positive integers (z + 1, x, y) can belong to a family of triplets of numbers, each member of which consists of a constant number z + 1 and two numbers x and at taking different values \u200b\u200bless than z + 1 ... Family members can be represented in the form of inequalities in which the constant left side is less, or more than the right side. Inequalities can be arranged in an orderly manner as a sequence of inequalities:
With a further increase in the exponent n\u003e k to infinity, none of the inequalities of the sequence (17) changes its meaning and turns into equality. In sequence (16), the inequality formed from an arbitrary triple of positive integers (z + 1, x, y) , can be on its right side in the form (z + 1) n\u003e x n + y n or be on its left side as (z + 1) n< x n + y n .
In any case, a triple of positive integers (z + 1, x, y) at n\u003e 2 , z + 1\u003e x , z + 1\u003e y in sequence (16) is an inequality and cannot represent an equality, i.e., it cannot represent a solution of the equation of the Great Fermat's 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 the opposite meaning. On the contrary, it is not easy and not easy 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), an increase in the integer degree of inequalities by 1 unit 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 the first power inequalities of the opposite meaning, there is necessarily a power equality. Its degree cannot be an integer, since there are only non-integers between two consecutive natural numbers. Power equality of a non-integer degree, according to the hypothesis 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, not a single left-side inequality remains, and only the right-side inequalities remain, which represent a sequence of increasing power inequalities (17). A further increase in their whole degree by 1 unit only strengthens its power inequalities and categorically excludes the possibility of the appearance of equality in a whole 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, as required.
Therefore, Fermat's Last Theorem is proved in all its universality:
- in section A) for all triples (z, x, y) Pythagorean numbers (Fermat's discovery is truly wonderful proof),
- in section B) for all family members 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.
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Changes were made on 05.09.2010. |
Which theorems can and cannot be proved by contradiction
In the explanatory dictionary of mathematical terms, a definition is given to a proof by contradiction of a theorem opposite to the inverse theorem.
“Proof by contradiction is a method of proving a theorem (proposition), which consists in proving not the theorem itself, but its equivalent (equivalent), opposite to the inverse (inverse to the opposite) theorem. A proof by contradiction is used whenever the direct theorem is difficult to prove, and the opposite is easier to prove. 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 the absurd and proves the theorem. "
Proof by contradiction is very common in mathematics. Proof by contradiction is based on the law of the excluded third, which is that of two statements (statements) A and A (negation 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 .: Education, 1965.- 539 p.
It would not be better to openly declare that the method of proving 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 a proof by contradiction "is used whenever the direct theorem is difficult to prove", when in fact it is used if and only if there is no substitute for it?
The characterization of the relationship of direct and inverse theorems to each other deserves special attention. “The converse theorem for a given theorem (or for 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 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, that is, 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 .: Education, 1965.- 539 p .: ill.-C.261 /
This characteristic of the relation between the direct and inverse theorem 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 converse 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 a logical method by contradiction.
Let us 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 form in a short form as follows: of AND should E ... Symbol AND the given condition of the theorem, accepted without proof, matters. Symbol E the meaning of the conclusion of the theorem that is required to be proved.
We will prove the direct theorem by contradiction, logical method. A 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 of AND should E , supplement with the opposite condition of AND it does not follow E .
The result is a logical contradictory condition of the new theorem, which contains two parts: of AND should E and of AND it does not follow 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 a contradictory condition is false, another part is true, and the third is excluded. Proof by contradiction has its task and aim 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 determined 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"... Evidence by contradiction there is reasoning during which it is established falsity (absurdity) of the conclusion arising from false conditions of the theorem being proved.
Given: of AND should E and from AND it does not follow E .
Prove: of AND should E .
Evidence: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its solution in the proof and its result. The result turns out to be false with flawless and error-free reasoning. With a logically correct reasoning, the reason for the false conclusion can only be a contradictory condition: of AND should E and of AND it does not follow E .
There is no shadow of a doubt that one part of the condition is false, while 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 found that would distinguish one part of the condition from another. Therefore, to the same extent it can be of AND should E and maybe of AND it does not follow E ... Statement of AND should E may be false, then the statement of AND it does not follow E will be true. Statement of AND it does not follow E may be false, then the statement of AND should E will be true.
Consequently, it is impossible to prove the direct theorem by contradiction.
Now we will prove the same direct theorem by the usual mathematical method.
Given: AND .
Prove: of AND should E .
Evidence.
1. Of AND should B
2. Of B should IN (by the previously proved theorem)).
3. Of IN should D (by the previously proved theorem).
4. Of D should D (by the previously proved theorem).
5. Of D should E (by the previously proved theorem).
Based on the law of transitivity, of AND should E ... The direct theorem is proved by the usual method.
Let the proved direct theorem have the correct converse theorem: of E should AND .
Let's prove it with the usual mathematical method. The proof of the converse theorem can be expressed symbolically as an algorithm of mathematical operations.
Given: E
Prove: of E should AND .
Evidence.
1. Of E should D
2. Of D should D (by the previously proved converse theorem).
3. Of D should IN (by the previously proved converse theorem).
4. Of IN it does not follow B (the converse theorem is not true). That's why of B it does not follow AND .
In this situation, it makes no sense to continue the mathematical proof of the converse theorem. The reason for the situation is logical. It is impossible to replace the incorrect converse theorem with anything. Therefore, it is impossible to prove this converse theorem by the usual mathematical method. All hope is for the proof of this converse theorem by the method of contradiction.
To prove it by contradictory method, it is required to replace its mathematical condition with a logical contradictory condition, which contains in its meaning two parts - false and true.
The converse theorem states: of E it does not follow AND ... Her condition E , from which follows the conclusion AND , is the result of proving the direct theorem by the usual mathematical method. This condition must be kept and supplemented with the statement of E should AND ... As a result of the addition, a contradictory condition of the new converse theorem is obtained: of E should AND and of E it does not follow AND ... Based on this logically contradictory condition, the converse theorem can be proved by means of the correct logical reasoning only, and only, logical by contradiction. In 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 of E should AND condition E was proved by the proof of the direct theorem. In the second part of E it does not follow AND condition E was assumed and accepted without proof. Some of them one is false and the other is true. It is required to prove which of them is false.
We prove by means of the correct logical reasoning and find that its result is a false, absurd conclusion. The reason for the false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. Only a statement can be a false part of E it does not follow AND , wherein E was accepted without proof. This is how it differs from E approval of E should AND , which is proved by the proof of the direct theorem.
Therefore, the following statement is true: of E should AND , as required to prove.
Conclusion: by a logical method, by contradiction, only that converse theorem is proved which has a direct theorem proved by a mathematical method and which cannot be proved by a mathematical method.
The obtained conclusion acquires an exceptional importance in relation to the method of proof by contradiction of the Great Fermat's 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 Wiles' Great Fermat Theorem is no exception.
Dmitry Abrarov in his article "Fermat's Theorem: The Phenomenon of Wiles' Proofs" published a commentary on the proof of the Great Fermat Theorem by Wiles. According to Abrarov, Wiles proves the Great Fermat's theorem with the help of a remarkable find by the German mathematician Gerhard Frey (b. 1944), who linked the potential solution of Fermat's equation x n + y n \u003d z n
where n\u003e 2
, with another, completely unlike it, equation. This new equation is given by a special curve (called the Frey elliptic curve). The Frey curve is given by an equation of a very simple form:
.
“Namely, Frey matched every solution (a, b, c) Fermat's equation, that is, numbers satisfying the relation a n + b n \u003d c nabove curve. In this case, the great Fermat's theorem would follow from here.(Quote from: Abrarov D. "Fermat's Theorem: The Wiles Evidence Phenomenon")
In other words, Gerhard Frey suggested that the equation of the great Fermat's theorem x n + y n \u003d z n
where n\u003e 2
, has solutions in positive integers. These solutions are, according to Frey's assumption, solutions of his equation
y 2 + x (x - a n) (y + b n) \u003d 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable find by Frey and with it through mathematical the method proved that this find, that is, the Frey elliptic curve, does not exist. Therefore, there is no equation and its solutions, which are given by a non-existent elliptic curve, Therefore, Wiles should have accepted the conclusion that the equation of the Great Fermat's theorem and Fermat's theorem itself do not exist. However, he made a more modest conclusion that the equation of the great Fermat's theorem has no solutions in positive integers.
It may be an irrefutable fact that Wiles accepted an assumption that is exactly the opposite in meaning to what is stated by Fermat's Last Theorem. It obliges Wiles to prove Fermat's Last Theorem by contradiction. We will follow his example and see what comes out of this example.
Fermat's Last Theorem states that the equation x n + y n \u003d z n where n\u003e 2 , has no solutions in positive integers.
According to the logical method of proof by contradiction, this statement is preserved, taken as given without proof, and then supplemented with the opposite statement in meaning: the equation x n + y n \u003d z n where n\u003e 2 , has solutions in positive integers.
The alleged 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, equal and equally possible. Through correct reasoning, it is required to establish which one is false, in order then to establish that the other statement is true.
The correct reasoning ends with a false, absurd conclusion, the logical reason for which can only be the contradictory condition of the theorem being proved, which contains two parts of the 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 found by which it could be established which particular statement is false. It could be the statement: the equation x n + y n \u003d z n where n\u003e 2 , has solutions in positive integers. On the same basis, it can be the statement: the equation x n + y n \u003d z n where n\u003e 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 proved by contradiction.
It would be a completely different matter if Fermat's Last Theorem were a converse theorem that has a direct theorem proven by the usual mathematical method. In this case, it could be proved by contradiction. And since it is a direct theorem, then its proof should be based not on the logical method of proving by contradiction, but on the usual mathematical method.
According to D. Abrarov, the most famous of modern Russian mathematicians, Academician V. I. Arnold, reacted to Wiles's proof "actively skeptically." The academician stated: “this is not real mathematics - real mathematics is geometric and strong in connection with physics.” (Quote from: Abrarov D. “Fermat's theorem: the phenomenon of Wiles's proofs.” The academician's statement expresses the very essence of Wiles's non-mathematical proof of the Great Fermat's theorem.
By contradiction it is impossible to prove either that the equation of the Great Fermat's theorem has no solutions, nor that it has solutions. Wiles's mistake is not mathematical, but logical - the use of proof by contradiction where its use does not make sense and does not prove the Great Fermat's theorem.
Fermat's Last Theorem is not proved using the usual mathematical method, if it is given: the equation x n + y n \u003d z n where n\u003e 2 , has no solutions in positive integers, and if it is required to prove in it: the equation x n + y n \u003d z n where n\u003e 2 , has no solutions in positive integers. In this form there is not a theorem, but a tautology devoid of meaning.
Note. My proof of BTF was discussed on one of the forums. One of Trotil's contributors, an expert in number theory, made the following authoritative statement entitled "A Brief Retelling of What Mirgorodsky Did." I quote it verbatim:
« AND. He proved that if z 2 \u003d x 2 + y then z n\u003e x n + y n ... This is a well-known and quite obvious fact.
IN. He took two triplets - Pythagorean and non-Pythagorean and showed by simple search that for a specific, specific family of triples (78 and 210 pieces), the BTF is fulfilled (and only for him).
FROM. And then the author omits the fact that < in a subsequent degree may be = , not only > ... A simple counterexample - transition n \u003d 1 in n \u003d 2 in the Pythagorean triplet.
D. This point does not contribute anything significant to the proof of BTF. Conclusion: BTF has not been proven. "
I will consider his conclusion point by point.
AND. It proved the BTF for the whole infinite set of triples of Pythagorean numbers. Proved by the geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was discovered, as I believe, by P. Fermat himself. This is what Fermat might have had in mind when he wrote:
"I have discovered a truly wonderful proof of this, but these fields are too narrow for him." This my assumption is based on the fact that in the Diophantus problem, against which, in the margins of the book, Fermat wrote, we are talking about solutions of the Diophantine equation, which are triples of Pythagorean numbers.
An infinite set of triples of Pythagorean numbers are solutions of the Diophatic 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. Later Fermat could extend his theorem to the set of all natural numbers. On the set of all natural numbers, the 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. At this point, I prove that both the family of an arbitrarily taken Pythagorean triplet of numbers and the family of an arbitrarily taken non-Pythagorean triplet of BTF numbers is satisfied.This is a necessary but insufficient and intermediate link in my proof of BTF. The examples I have taken of a family of a triple of Pythagorean numbers and a family of a triple of non-Pythagorean numbers have the meaning of specific examples that assume and do not exclude the existence of similar other examples.
Trotil's assertion that I “showed by a simple search that for a specific, definite family of triplets (78 and 210 pieces), the BTF is fulfilled (and only for it) is without foundation. He cannot refute the fact that I can just as well take other examples of the Pythagorean and non-Pythagorean triplets to obtain a specific specific family of one and the other triplets.
Whichever pair of triplets I take, their suitability for solving the problem can be checked, in my opinion, only by the “simple search” method. Some other method is not known to me and is not required. If Trotil doesn't like it, then he should have suggested another method, which he doesn't. Without offering anything in return, it is incorrect to condemn “simple brute force”, which in this case is irreplaceable.
FROM. I omitted \u003d between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 \u003d x 2 + y (1), in which the degree n\u003e 2 — whole positive number. From the equality between the inequalities it follows obligatory consideration of equation (1) for non-integer degree n\u003e 2 ... Trotil counting compulsory considering equality between inequalities, in fact considers necessary in the proof of the BTF, consideration of Eq. (1) for incomplete the meaning of the degree n\u003e 2 ... I did this for myself and found that equation (1) for incomplete the meaning of the degree n\u003e 2 has a solution of three numbers: z, (z-1), (z-1) with a non-integer exponent.
In the last twentieth century, an event occurred on a scale of equal scale in mathematics in its entire history. On September 19, 1994, the theorem formulated by Pierre de Fermat (1601-1665) more than 350 years ago in 1637 was proved. It is also known as "Fermat's last theorem" or as "Fermat's last theorem", since there is also the so-called "Fermat's little theorem". It was proved by a 41-year-old professor of Princeton University, Andrew Wiles, who was not particularly remarkable in the mathematical community until this moment, and by mathematical standards is no longer young.
It is surprising that not only our ordinary Russian inhabitants, but also many people interested in science, including even a considerable number of scientists in Russia, who use mathematics in one way or another, do not really know about this event. This is shown by the never-ending "sensational" reports of "elementary proofs" of Fermat's theorem in popular Russian newspapers and on television. The next evidence was covered with such informational power, as if the most authoritative examination and the most widely known throughout the world Wiles's proof did not exist. The response of the Russian mathematical community to this front-page news has been surprisingly sluggish in the face of long-standing rigorous proof. Our goal is to sketch the gripping and dramatic story of Wiles's proof in the context of the enchanting story of Fermat's greatest theorem, and talk a little about the proof itself. Here we are primarily interested in the question of the possibility of an accessible presentation of Wiles's proof, which, of course, most mathematicians in the world know about, but only very, very few of them can talk about understanding this proof.
So, let's recall the famous Fermat's theorem. Most of us have heard about her in one way or another since school days. This theorem is related to a very significant equation. This is perhaps the simplest meaningful equation you can write using three unknowns and another strictly positive integer parameter. Here it is:
Fermat's Last Theorem states that for values \u200b\u200bof the parameter (degree of the equation) exceeding two, there are no integer solutions of this equation (except, of course, a solution when all these variables are equal to zero simultaneously).
The attractive power of this Fermat's theorem for the general public is obvious: there is no other mathematical statement that has such a simplicity of formulation, the seeming availability of proof, as well as the attractiveness of its "status" in the eyes of society.
Before Wiles, an additional incentive for fermatists (the so-called people who obsessively attacked Fermat's problem) was the German Wolfskehl Prize for Proof, established almost a hundred years ago, though small compared to the Nobel Prize - it managed to depreciate during the First World War.
In addition, she was always attracted by the probable elementary nature of the proof, since Fermat himself "proved it", writing in the margin of Diophantus' translation of Arithmetic: “I found a truly wonderful proof of this, but the margins here are too narrow to accommodate it”.
That is why it is appropriate here to give an assessment of the relevance of the popularization of Wiles's proof of Fermat's problem, due to the famous American mathematician R. Murty (we quote from the forthcoming translation of the book by Y. Manin and A. Panchishkin "Introduction to modern number theory"):
“Fermat's Great Theorem occupies a special place in the history of civilization. With its outward simplicity, it has always attracted both amateurs and professionals ... Everything looks as if it had been conceived by some higher mind, which for centuries has developed various directions of thought only to then reunite them into one exciting alloy for solving the Big Fermat's theorem. No human can claim to be an expert on all the ideas used in this "miraculous" proof. In an era of universal specialization, when each of us knows "more and more about less and less", it is absolutely necessary to have an overview of this masterpiece ... "
Let's start with a brief historical background, largely inspired by Simon Singh's fascinating book, Fermat's Last Theorem. Serious passions always boiled around the insidious theorem, alluring in its seeming simplicity. The history of its proof is continuous drama, mysticism and even direct victims. Perhaps the most iconic victim is Yutaka Taniyama (1927-1958). It was this young, talented Japanese mathematician, who was distinguished in his life by great extravagance, who created the basis for Wiles' attack in 1955. On the basis of his ideas, Goro Shimura and André Weil a few years later (60-67 years) finally formulated the famous hypothesis, having proved a significant part of which, Wiles obtained Fermat's theorem as a consequence. The mysticism of the story of the death of a non-trivial Yutaka is associated with his violent temperament: he hanged himself at the age of thirty-one out of unhappy love.
The whole long history of the mysterious theorem has been accompanied by constant announcements of its proof, starting with Fermat himself. Constantly found errors in an endless stream of proofs comprehended not only amateur mathematicians, but also professional mathematicians. This led to the fact that the term "fermatist", applied to those who prove Fermat's theorem, became a household name. The everlasting intrigue with its proof sometimes led to amusing incidents. So, when a gap was discovered in the first version of Wiles's already widely advertised proof, a snide inscription appeared on one of the New York subway stations: "I found a truly wonderful proof of Fermat's Last Theorem, but my train came and I did not have time to write it down."
Andrew Wiles, born in England in 1953, studied mathematics at Cambridge; in graduate school was with Professor John Coates. Under his leadership, Andrew comprehended the theory of the Japanese mathematician Iwasawa, located on the border of classical number theory and modern algebraic geometry. This fusion of seemingly distant mathematical disciplines is called arithmetic algebraic geometry. Andrew challenged Fermat's problem, relying on this synthetic theory, which is complex even for many professional mathematicians.
After graduating from graduate school, Wiles received a position at Princeton University, where he still works. He is married and has three daughters, two of whom were born "in a seven-year first-draft trial." During these years, only Nada, Andrew's wife, knew that he alone was storming the most inaccessible and most famous summit of mathematics. It is to them, Nadia, Claire, Kate and Olivia, that Wiles' famous final article "Modular Elliptic Curves and Fermat's Last Theorem" is dedicated to the central mathematical journal "Annals of Mathematics", where the most important mathematical works are published.
The events themselves around the evidence unfolded rather dramatically. This exciting scenario could be called a "fermatist-professional mathematician."
Indeed, Andrew had dreamed of proving Fermat's theorem since his youth. But, unlike the overwhelming majority of fermatists, it was clear to him that this requires mastering whole layers of the most complex mathematics. Moving towards his goal, Andrew graduated from the mathematics department of the famous University of Cambridge and began to specialize in modern number theory, which is at the interface with algebraic geometry.
The idea of \u200b\u200bstorming the shining peak is quite simple and fundamental - the best possible ammunition and careful planning of the route.
As a powerful tool for achieving the goal, the already familiar Iwasawa theory developed by Wiles himself, which has deep historical roots, is chosen. This theory generalized Kummer's theory - historically the first serious mathematical theory to storm Fermat's problem, which appeared back in the 19th century. In turn, the roots of Kummer's theory lie in the famous theory of the legendary and brilliant romantic revolutionary Evariste Galois, who died at the age of twenty-one in a duel in defense of the honor of a girl (note, remembering the story with Taniyama, on the fatal role of beautiful ladies in the history of mathematics) ...
Wiles is completely immersed in proof, even ceasing to participate in scientific conferences. And as a result of seven years of hermitage from the mathematical community in Princeton, in May 1993, Andrew puts an end to his text - it's done.
It was at this time that an excellent reason was tucked up to notify the scientific world about your discovery - in June, a conference was to be held in his native Cambridge on the right topic. Three lectures at the Isaac Newton Institute in Cambridge excite not only the mathematical world, but also the general public. At the end of the third lecture, on June 23, 1993, Wiles announces the proof of Fermat's Last Theorem. The proof is full of a whole bunch of new ideas, such as a new approach to the Taniyama-Shimura-Weil conjecture, the far advanced Iwasawa theory, and the new "strain control theory" of Galois representations. The mathematical community is eagerly awaiting proof text by experts in arithmetic algebraic geometry.
This is where that very dramatic turn comes. Wiles himself, in the process of communicating with reviewers, discovers a gap in his proof. The crack was created by the mechanism of "deformation control" invented by him - the supporting structure of the proof.
The gap is discovered a couple of months later by Wiles' concise explanation of his proof to Princeton department colleague Nick Katz. Nick Katz, having been on friendly terms with Andrew for a long time, recommends that he cooperate with the young promising English mathematician Richard Taylor.
Another year of hard work is going on, connected with the study of an additional weapon of attack on an intractable problem - the so-called Euler systems, independently discovered in the 1980s by our compatriot Viktor Kolyvagin (who has long worked at New York University) and Thane.
And here is a new test. The incomplete, but still very impressive result of Wiles's work is reported by him to the International Congress of Mathematicians in Zurich at the end of August 1994. Wiles fights with all his might. Literally before the report, according to eyewitnesses, he is still feverishly writing something, trying to maximize the situation with the "sagging" evidence.
After this intriguing audience of the world's largest mathematicians, Wiles' speech, the mathematical community "exhales joyfully" and sympathetically applauds: nothing, guy, it happens to anyone, but he advanced science, showing that in solving such an unapproachable hypothesis one can successfully advance, which no one has ever done before did not even think of doing. Yet another fermatist, Andrew Wiles, could not take away the cherished dream of many mathematicians to prove Fermat's theorem.
It is natural to imagine Wiles's condition at the time. Even the support and friendly attitude of his colleagues could not compensate for his state of psychological devastation.
And so, just a month later, when, as Wiles writes in the introduction to his final article in the Annals with the definitive proof, “I decided to give up my last glance at Euler systems in an attempt to reanimate this argument for proof,” it happened. A flash of insight hit Wiles on September 19, 1994. It was on that day that the gap in the evidence was closed.
Then things went at a rapid pace. The already established cooperation with Richard Taylor in the study of the Euler systems of Kolyvagin and Thane allowed the proof to be finalized in the form of two large papers already in October.
Their publication, which occupied the entire issue of "Annals of Mathematics", followed in November 1994. All this caused a new powerful information surge. The story of Wiles' proof received rave press in the United States, a film was made and books were published about the author of a fantastic breakthrough in mathematics. In one assessment of his own work, Wiles noted that he had invented the mathematics of the future.
(I wonder if this is so? Let us only note that with all this information flurry the practically zero information resonance in Russia, which continues to this day, was in sharp contrast).
Let us ask ourselves a question - what is the “inner kitchen” of obtaining outstanding results? After all, it is interesting to know how a scientist organizes his work, what he is guided by, how he determines the priorities of his activities. What can be said in this sense about Andrew Wiles? And suddenly it turns out that in the modern era of active scientific communication and collective working style, Wiles had his own view of the style of working on superproblems.
Wiles went to his fantastic result through intense, continuous, individual work. The organization of his activities, in official language, was extremely unplanned. This categorically could not be called an activity within the framework of a certain grant, on which it is necessary to regularly report and again, each time, plan to obtain certain results by a certain date.
Such activities outside of society, which did not use direct scientific communication with colleagues even at conferences, seemed to contradict all the canons of the work of a modern scientist.
But it was the individual work that made it possible to go beyond the already established standard concepts and methods. This style of work, closed in form and at the same time free in essence, made it possible to invent powerful new methods and obtain results of a new level.
The problem facing Wiles (the Taniyama-Shimura-Weil conjecture) was not even among the nearest peaks in those years that can be conquered by modern mathematics. At the same time, none of the experts denied its enormous importance, and nominally, it was in the "mainstream" of modern mathematics.
Thus, Wiles's activities were of a pronounced non-systemic nature and the result was achieved thanks to the strongest motivation, talent, creative freedom, will, more than favorable material conditions for working at Princeton and, which is extremely important, mutual understanding in the family.
Wiles' proof, which appeared like a bolt from the blue, has become a kind of test for the international mathematical community. The reaction of even the most progressive part of this community as a whole turned out, oddly enough, rather neutral. After the emotions and enthusiasm subsided for the first time after the appearance of the landmark evidence, everyone calmly continued their business. Specialists in arithmetic algebraic geometry slowly studied the "mighty proof" in their narrow circle, while the rest plowed their mathematical paths, diverging, as before, further and further from each other.
Let's try to understand this situation, which has both objective and subjective reasons. Objective factors of non-perception, oddly enough, have roots in the organizational structure of modern scientific activity. This activity is like a skating rink going down a sloping road and having colossal momentum: its own school, its established priorities, its sources of funding, etc. All this is good from the point of view of an established system of reporting to the grantor, but it makes it difficult to raise your head and look around: what is actually really important and relevant for science and society, and not for the next portion of the grant?
Then - again - I don't want to crawl out of my cozy burrow, where everything is so familiar, and climb into another, completely unfamiliar hole. It is not known what to expect there. Moreover, it is obviously clear that they don’t give money for the invasion.
It is quite natural that none of the bureaucratic structures organizing science in different countries, including Russia, did not draw conclusions not only from the phenomenon of Andrew Wiles' proof, but also from the similar phenomenon of the sensational proof of Grigory Perelman of another, also famous mathematical problem.
The subjective factors of neutrality of the reaction of the mathematical world to the "event of the millennium" lie in quite prosaic reasons. The proof is indeed extraordinarily complex and long. To the layman in arithmetic algebraic geometry, it seems to consist of a layering of terminology and constructions from the most abstract mathematical disciplines. It seems that the author did not at all set a goal to be understood by as many interested mathematicians as possible.
This methodological complexity, unfortunately, is present as an inevitable cost of the great evidence of recent times (for example, the analysis of the recent proof of the Poincaré hypothesis by Grigory Perelman continues to this day).
The complexity of perception is further enhanced by the fact that arithmetic algebraic geometry is a very exotic subdomain of mathematics, which causes difficulties even for professional mathematicians. The matter was also aggravated by the extraordinary synthetic nature of Wiles's proof, which used a variety of modern tools, created by a large number of mathematicians in recent years.
But we must take into account that Wiles was not faced with the methodical task of explanation - he designed a new method. It was the synthesis of Wiles' own ingenious ideas and a conglomerate of the latest results from various mathematical directions that worked in the method. And it was such a powerful structure that rammed an unapproachable problem. The proof was no accident. The fact of its crystallization fully corresponded to both the logic of the development of science and the logic of cognition. The task of explaining this super-proof seems to be an absolutely independent, rather difficult, albeit very promising problem.
You can probe public opinion for yourself. Try asking some mathematicians you know about Wiles' proof: who understood? Who understood even the basic ideas? Who wanted to understand? Who felt that this was new mathematics? The answers to these questions seem rhetorical. And you are unlikely to meet many people who want to break through the palisade of technical terms and master new concepts and methods in order to solve just one very exotic equation. And why is it necessary to study all this for the sake of this particular task ?!
Let me give you a funny example. A couple of years ago, the famous French mathematician, Fields laureate, Pierre Deligne, a leading specialist in algebraic geometry and number theory, when asked by the author about the meaning of one of the key objects of Wiles's proof - the so-called "deformation ring" - after half an hour of thought, said that he was not completely understands the meaning of this object. It has been ten years since the proof to this point.
Now you can reproduce the reaction of Russian mathematicians. The main reaction is its almost complete absence. This is mainly due to Wiles's "heavy" and "unusual" math.
For example, in classical number theory you will not find such lengthy proofs as in Wiles. As number theorists put it, "the proof must be on a page" (Wiles's proof, in collaboration with Taylor, is 120 pages long in the magazine version).
Also, the factor of fear for the lack of professionalism of your assessment cannot be ruled out: by reacting, you take responsibility for assessing the evidence. And how to do it when you don't know this math?
The position taken by direct specialists in number theory is characteristic: "... and awe, and burning interest, and caution in the face of one of the greatest mysteries in the history of mathematics" (from the preface to Paulo Ribenboim's book "Fermat's Last Theorem for Amateurs" - the only one available today day to source directly from Wiles' proof for the general reader.
The reaction of one of the most famous contemporary Russian mathematicians, academician V.I. Arnold's proof is "actively skeptical": this is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its nature, cannot generate the development of mathematics, since it is “binary”, that is, the formulation of the problem requires an answer only to the question “yes or no”. At the same time, the mathematical work of recent years by V.I. Arnold were largely devoted to variations on a very similar number-theoretic topic. It is possible that Wiles paradoxically became an indirect cause of this activity.
At the Faculty of Mechanics and Mathematics of Moscow State University, nevertheless, enthusiasts of proof appear. The remarkable mathematician and popularizing scientist Yu.P. Soloviev (who left us untimely) initiates the translation of E. Knapp's book on elliptic curves with the necessary material on the Taniyama – Shimura – Weil hypothesis. Alexei Panchishkin, now working in France, in 2001 reads lectures on mechanics and mathematics, which formed the basis of the corresponding part of it with Yu.I. Manin's excellent book on modern number theory mentioned above (published in Russian translation by Sergei Gorchinsky and edited by Aleksey Parshin in 2007).
It is somewhat surprising that at the Steklov Mathematical Institute in Moscow, the center of the mathematical world in Russia, Wiles's proof was not examined at seminars, but was studied only by individual specialized experts. Moreover, the proof of the already complete Taniyama-Shimura-Weil conjecture was not understood (Wiles proved only a part of it sufficient to prove Fermat's theorem). This proof was given in 2000 by a whole group of foreign mathematicians, including Richard Taylor, Wiles's co-author on the final stage of the proof of Fermat's theorem.
Also, there were no public statements and, moreover, discussions on the part of well-known Russian mathematicians about Wiles's proof. There is a rather sharp discussion between the Russian V. Arnold ("the skeptic of the method of proof") and the American S. Leng ("the enthusiast of the method of proof"), however, its traces are lost in Western publications. In the Russian central mathematical press since the publication of Wiles's proof, there have been no publications on the topic of proof. Perhaps the only publication on this topic was the translation of an article by the Canadian mathematician Henry Darmon of even an inconclusive version of the proof in Successes of Mathematical Sciences in 1995 (funny that the complete proof has already been published).
Against this "sleepy" mathematical background, despite the extremely abstract nature of Wiles's proof, some fearless theoretical physicists included it in their area of \u200b\u200bpotential interest and began to study it, hoping sooner or later to find applications of Wiles's mathematics. This cannot but rejoice, if only because this mathematics has been practically in self-isolation all these years.
Nevertheless, the problem of proof adaptation, which greatly aggravates its applied potential, remained and remains very urgent. To date, the original highly specialized text of Wiles' paper and the joint paper by Wiles and Taylor have already been adapted, albeit only for a fairly narrow circle of professional mathematicians. This is done in the aforementioned book by Yu. Manin and A. Panchishkin. They have successfully ironed out a certain artificiality of the original proof. In addition, the American mathematician Serge Lang, a fierce propagandist of Wiles' proof (unfortunately left us in September 2005), included some of the most important proof constructs in the third edition of his now classic university textbook Algebra.
As an example of the artificiality of the original proof, we note that one of the particularly striking features that create this impression is the special role of individual primes, such as 2, 3, 5, 11, 17, as well as individual natural numbers, such as 15, 30, and 60. Among other things, it is quite obvious that the proof is not geometric in the most ordinary sense. It does not contain natural geometric images to which one could attach for a better understanding of the text. Heavy-duty "terminated" abstract algebra and "advanced" number theory purely psychologically beat the perception of proof even for a qualified mathematician reader.
One can only wonder why, in such a situation, the experts of proof, including Wiles himself, do not “polish” him, do not propagandize and popularize the obvious “mathematical hit” even in his native mathematical community.
So, to put it briefly, today the fact of Wiles' proof is simply the fact of proving Fermat's theorem with the status of the first correct proof and used in it "some super-powerful mathematics".
The well-known Russian mathematician of the middle of the last century, the former dean of the Faculty of Mathematics, V.V. Golubev:
“... according to the witty remark of F. Klein, many departments of mathematics are similar to those exhibitions of the latest models of weapons that exist at the firms that manufacture weapons; with all the wit put by the inventors, it often happens that when a real war breaks out, these novelties turn out to be unsuitable for one reason or another ... Exactly the same picture is presented by the modern teaching of mathematics; students are given very perfect and powerful means of mathematical research in their hands ..., but then the students cannot stand any idea of \u200b\u200bwhere and how these powerful and ingenious methods can be applied in solving the main problem of all science: in knowing the world around us and in influencing him the creative will of man. At one time A.P. Chekhov said that if in the first act of the play a gun hangs on the stage, then it is necessary that at least in the third act they shoot from it. This remark is fully applicable to teaching mathematics: if a theory is presented to students, then sooner or later it is necessary to show what applications can be made from this theory, primarily in the field of mechanics, physics or technology and in other areas.
Continuing this analogy, we can say that Wiles's proof is extremely favorable material for studying a huge layer of modern fundamental mathematics. Here students can be shown how the problem of classical number theory is closely related to such branches of pure mathematics as modern algebraic number theory, modern Galois theory, p-adic mathematics, arithmetic algebraic geometry, commutative and non-commutative algebra.
It would be fair if Wiles' confidence that the mathematics he invented - mathematics of a new level - was confirmed. And I really do not want this really very beautiful and synthetic mathematics to suffer the fate of a “gun that never fired”.
And yet, let us now ask ourselves the question: is it possible to describe Wiles's proof in sufficiently accessible terms for a wide interested audience?
From the point of view of specialists, this is an absolute utopia. But let's, all the same, try, guided by a simple argument that Fermat's theorem is a statement only about the whole points of our usual three-dimensional Euclidean space.
We will successively substitute points with integer coordinates into Fermat's equation.
Wiles finds an optimal mechanism for recalculating integer points and testing them to satisfy the equation of Fermat's theorem (after introducing the necessary definitions, such recalculation will exactly correspond to the so-called "property of modularity of elliptic curves over the field of rational numbers" described by the Taniyama – Shimura – Weil conjecture ”).
The recalculation mechanism is optimized with the help of a remarkable find by the German mathematician Gerhard Frey, who linked the potential solution of Fermat's equation with an arbitrary exponent to another, completely different from it, equation. This new equation is given by a special curve (called the Frey elliptic curve). This Frey curve is given by an equation of a very simple form:
The unexpectedness of Frey's idea consisted in the transition from the number-theoretic nature of the problem to its "hidden" geometric aspect. Namely: Frey compared to every solution of Fermat's equation, that is, to numbers satisfying the relation
the above curve. Now it remained to show that such curves do not exist for. In this case, the great Fermat's theorem would follow from here. It was this strategy that was chosen by Wiles in 1986, when he began his enchanting assault.
Frey's invention at the time of Wiles's "start" was quite fresh (85th year) and also echoed the relatively recent approach of the French mathematician Helleguarsh (70s), who proposed using elliptic curves to find solutions to Diophantine equations, i.e. equations similar to Fermat's equation.
Let's now try to look at the Frey curve from a different point of view, namely, as a tool for recalculating integer points in Euclidean space. In other words, for us the Frey curve will play the role of a formula that determines the algorithm for such a recalculation.
In this context, we can say that Wiles invents tools (special algebraic constructions) to control this recalculation. As a matter of fact, it is Wiles' subtle toolkit that constitutes the central core and the main difficulty of the proof. It is in the manufacture of these instruments that Wiles' main sophisticated algebraic discoveries, which are so difficult to perceive, arise.
But nevertheless, the most unexpected effect of the proof is, perhaps, the sufficiency of using only one "Frey" curve, which is represented by a completely uncomplicated, almost "school" dependence. Surprisingly, the use of only one such curve turns out to be sufficient to test all points of the three-dimensional Euclidean space with integer coordinates to satisfy their relation to the Great Fermat's Theorem with an arbitrary exponent.
In other words, the use of just one curve (albeit with a specific form), understandable even for an ordinary high school student, turns out to be tantamount to building an algorithm (program) for sequential recalculation of integer points in ordinary three-dimensional space. And not just a recalculation, but a recalculation with simultaneous testing of an integer point for “its satisfaction” with Fermat's equation.
It is here that the phantom of Pierre de Fermat himself arises, since this recalculation brings to life what is usually called “Ferma’t descent”, or Fermat’s reduction (or “method of infinite descent”).
In this context, it immediately becomes clear why Fermat himself could not prove his theorem for objective reasons, although at the same time he could well “see” the geometric idea of \u200b\u200bits proof.
The fact is that the recalculation is carried out according to the control of mathematical instruments that have no analogues not only in the distant past, but also unknown before Wiles even in modern mathematics.
The most important thing here is that these tools are "minimal", i.e. they cannot be simplified. Although this “minimalism” in itself is very difficult. And it was Wiles' awareness of this non-trivial “minimality” that became the decisive final step in the proof. This was exactly the same "flash" on September 19, 1994.
Some problem that causes dissatisfaction still remains here - Wiles does not explicitly describe this minimal construction. Therefore, those interested in Fermat's problem still have interesting work - a clear interpretation of this "minimality" is needed.
It is possible that this is where the geometry of the "over-algebraized" proof should be hidden. It is possible that Fermat himself felt this very geometry when he made the famous entry in the narrow margins of his treatise: "I found a truly remarkable proof ...".
Now let's go directly to the virtual experiment and try to "delve into" the thoughts of the mathematician-lawyer Pierre de Fermat.
The geometric image of the so-called Little Fermat's theorem can be represented as a circle rolling "without slipping" along a straight line and "winding" whole points on itself. In this interpretation, the equation of Fermat's little theorem also acquires a physical meaning - the meaning of the law of conservation of such motion in one-dimensional discrete time.
These geometric and physical images can be tried to transfer to a situation when the dimension of the problem (the number of variables in the equation) increases and the equation of the Little Fermat's theorem turns into the equation of the Big Fermat's theorem. Namely: let us assume that the geometry of the Great Fermat's theorem is represented by a sphere rolling on a plane and "winding" whole points on this plane. It is important that this rolling should not be arbitrary, but "periodic" (mathematicians also say "cyclotomic"). The rolling periodicity means that the vectors of the linear and angular velocity of the sphere rolling in the most general way after a certain fixed time (period) are repeated in magnitude and direction. This periodicity is analogous to the periodicity of the linear speed of a circle rolling along a straight line that simulates the "small" Fermat equation.
Accordingly, Fermat's "big" equation acquires the meaning of the conservation law of the above-mentioned motion of the sphere already in two-dimensional discrete time. Let us now take the diagonal of this two-dimensional time (this is the step in which the whole difficulty lies!). This extremely tricky and turns out to be the only diagonal is the equation of the great Fermat theorem, when the exponent of the equation is exactly two.
It is important to note that in a one-dimensional situation - the situation of Fermat's little theorem - there is no need to find such a diagonal, since the time is one-dimensional and there is nothing to take the diagonal for. Therefore, the degree of the variable in the equation of Fermat's little theorem can be arbitrary.
So, quite unexpectedly, we get a bridge to the "officialization" of Fermat's great theorem, that is, to the appearance of its physical meaning. How can one fail to remember that Fermat was not a stranger to physics.
By the way, the experience of physics also shows that the conservation laws of mechanical systems of the above form are quadratic in the physical variables of the problem. And finally, all this is quite consistent with the quadratic structure of the energy conservation laws of Newtonian mechanics, known from the school.
From the point of view of the above "physical" interpretation of the Great Fermat's theorem, the property of "minimality" corresponds to the minimality of the degree of the conservation law (this is two). And the reduction of Fermat and Wiles corresponds to the reduction of the conservation laws for recounting points to a law of the simplest form. This simplest (minimal in complexity) recalculation, both geometrically and algebraically, is represented by the rolling of a sphere on a plane, since a sphere and a plane are, as we completely understand, “minimal” two-dimensional geometric objects.
The whole difficulty, which at first glance is absent, is that the exact description of such a seemingly "simple" motion of the sphere is not at all easy. The point is that the "periodic" rolling of the sphere "absorbs" a bunch of so-called "hidden" symmetries of our three-dimensional space. These hidden symmetries are due to non-trivial combinations (compositions) of linear and angular motion of the sphere - see Fig. 1.
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It is precisely for an accurate description of these hidden symmetries, geometrically encoded by such a cunning rolling of the sphere (points with integer coordinates "sit" at the nodes of the drawn lattice) that Wiles's algebraic constructions are required.
In the geometric interpretation shown in Fig. 1, the linear motion of the center of the sphere "counts" whole points on the plane, and its angular (or rotational) motion provides the spatial (or vertical) component of the recalculation. It is not immediately possible to "see" the rotational motion of the sphere in the arbitrary rolling of the sphere on the plane. It is the rotational motion that corresponds to the above-mentioned hidden symmetries of Euclidean space.
The Frey curve introduced above just "encodes" the most beautiful from the aesthetic point of view, recalculation of whole points in space, reminiscent of movement along a spiral staircase. Indeed, if you follow the curve swept out by a certain point of the sphere in one period, you will find that our marked point sweeps out the curve shown in Fig. 2, which resembles a "double spatial sinusoid" - a spatial analogue of the graph. This nice curve can be interpreted as a graph of the "minimum" along (that is) Frey's curve. This is the graph of our testing recalculation.
Having connected some associative perception of this picture, we will discover to our surprise that the surface bounded by our curve is strikingly similar to the surface of a DNA molecule - the "cornerstone" of biology! It is perhaps no coincidence that the terminology of the DNA-encoding of constructs from Wiles's proof is used in Singh's book "Fermat's Last Theorem".
We emphasize once again that the decisive moment of our interpretation is the fact that an analog of the conservation law for the small Fermat theorem (its degree can be arbitrarily large) turns out to be the equation of the Great Fermat's theorem precisely in the case. It is this effect of “the minimality of the degree of the law of conservation of the rolling of a sphere on a plane” and corresponds to the statement of Fermat's Last Theorem.
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It is possible that Fermat himself saw or felt these geometric and physical images, but at the same time he could not assume that they are so difficult to describe from a mathematical point of view. Moreover, he could not assume that to describe such, albeit non-trivial, but still quite transparent geometry, it would take another three hundred and fifty years of work of the mathematical community.
Now let's throw a bridge to modern physics. The geometrical image of Wiles' proof proposed here is very close to the geometry of modern physics, trying to get close to the mystery of the nature of gravity - quantum general relativity. To confirm this, at first sight unexpected, interaction of Fermat's Last Theorem and "Big Physics", let us imagine that the rolling sphere is massive and "pushes" the plane underneath. The interpretation of this "punching" in Fig. 3 strikingly resembles the well-known geometric interpretation of Einstein's general theory of relativity, which describes precisely the "geometry of gravity."
And if we take into account also the present discretization of our picture, embodied by a discrete integer lattice on the plane, then we are witnessing "quantum gravity" with our own eyes!
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It is on this major "unifying" physical and mathematical note that we will end our "cavalry" attempt to give a visual interpretation of Wiles's "over-abstract" proof.
Now, perhaps, it should be emphasized that in any case, whatever the correct proof of Fermat's theorem, it must in one way or another use the constructions and logic of Wiles's proof. It is simply impossible to get around all this because of the aforementioned "minimality property" of Wiles's mathematical tools used for the proof. In our "geometrical-dynamic" interpretation of this proof, this "minimality property" provides the "minimum necessary conditions" for the correct (ie "convergent") construction of the testing algorithm.
On the one hand, this is a huge grief for amateurs-fermatists (if, of course, they find out about it; as they say, “the less you know, the better you sleep”). On the other hand, the natural "irreducibility" of Wiles's proof formally makes life easier for professional mathematicians - they may not read recurring "elementary" proofs from amateurs of mathematics, citing the lack of correspondence with Wiles's proof.
The general conclusion is that both of them need to "strain" and understand this "fanatical" proof, comprehending in essence "all mathematics".
What else is important not to miss when summing up this unique story that we have witnessed? The strength of Wiles's proof is that it is not just a formal logical reasoning, but a broad and powerful method. This creation is not a separate tool for proving one single result, but an excellent set of well-chosen tools that allows you to "split" a wide variety of problems. It is also fundamentally important that if we look down from the height of the skyscraper of Wiles's proof, we will see all the preceding mathematics. Paphos lies in the fact that it will not be "patchwork", but panoramic vision. All this speaks not only of the scientific, but also of the methodological continuity of this truly magical proof. There is nothing left - just to understand it and learn to apply it.
I wonder what our contemporary hero Wiles is doing today? There is no special news about Andrew. He, of course, received various awards and prizes, including the famous Wolfskel Prize, which was depreciated during the first civil war. For all the time that has passed since the triumph of the proof of Fermat's problem to the present day, I have managed to notice only one, albeit as always large, article in the same “Annals” (co-authored with Skinner). Maybe Andrew is hiding again on the eve of a new mathematical breakthrough, for example, the so-called “abc” hypothesis - recently formulated (by Masser and Osterle in 1986) and considered the most important problem in number theory today (this is the “problem of the century” as Serge Lang put it ).
Much more information about Wiles' co-author on the final part of the proof - Richard Taylor. He was one of four authors of the proof of the complete Taniyama-Shmura-Weil conjecture and was a serious candidate for the Fields Medal at the 2002 China Mathematical Congress. However, he did not receive it (then only two mathematicians got it - the Russian mathematician from Princeton Vladimir Voevodsky "for the theory of motives" and the Frenchman Laurent Laforgue "for an important part of Langlands' program"). Taylor has published a considerable number of remarkable works during this time. And just recently, Richard achieved a new great success - he proved a very famous conjecture - the Tate-Saito conjecture, also related to arithmetic algebraic geometry and generalizing the results of German. the 19th century mathematician G. Frobenius and the 20th century Russian mathematician N. Chebotarev.
Let's have a little fantasy in the end. Perhaps the time will come when mathematics courses in universities, and even in schools, will be tailored to Wiles's methods of proof. This means that Fermat's Last Theorem will become not only a model mathematical problem, but also a methodological model for teaching mathematics. On her example, it will be possible to study, in fact, all the main sections of mathematics. Moreover, future physics, and maybe even biology and economics, will begin to rely on this mathematical apparatus. But what if?
It seems that the first steps in this direction have already been taken. This is evidenced, for example, by the fact that the American mathematician Serge Lang included in the third edition of his classic manual on algebra the basic constructions of Wiles' proof. The Russian Yuri Manin and Alexei Panchishkin go even further in the aforementioned new edition of their Modern Number Theory, setting out in detail the proof itself in the context of modern mathematics.
And how not to exclaim now: Fermat's great theorem "died" - long live the Wiles method!
There are not so many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only mathematical problem that has received such wide popularity and has become a real legend. It is mentioned in many books and films, while the main context of almost all references is the impossibility of proving the theorem.
Yes, this theorem is very famous and, in a sense, has become an "idol" worshiped by amateur mathematicians and professionals, but few people know that its proof has been found, and it happened back in 1995. But first things first.
So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in its essence and understandable to any person with secondary education. It says that the formula a to the power n + b to the power n \u003d c to the power n has no natural (that is, non-fractional) solutions for n\u003e 2. It seems that everything is simple and clear, but the best mathematicians and ordinary amateurs fought over seeking a solution for more than three and a half centuries.
Why is she so famous? We'll find out now ...
Are there few proven, unproven, and not yet proven theorems? The point is that Fermat's Last Theorem is the greatest contrast between simplicity of formulation and complexity of proof. Fermat's Last Theorem is an incredibly difficult task, and nevertheless, its formulation can be understood by everyone with the 5th grade of high school, but the proof is not even by every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics, there is not a single problem that would be formulated so simply, but remained unsolved for so long. 2. What does it consist of?
Let's start with the 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-angled 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 triples of integers satisfying the equality x² + y² \u003d z². They proved that there are infinitely many Pythagorean triplets, and received general formulas for finding them. Probably, they tried to look for triples and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.
That is, it is easy to find a set of numbers that perfectly satisfy the equality x² + y² \u003d z²
Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 \u003d 25.
Or 5, 12, 13: 25 + 144 \u003d 169. Great.
So, it turns out that they are NOT. This is where the catch begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, you can and should just give this solution.
Proving the absence is more difficult: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - here it is, the solution! (please provide a solution). And that's it, the opponent is slain. How to prove absence?
Say, "I haven't found such solutions"? Or maybe you were looking badly? What if they are, only very large, well, very, such that even a super-powerful computer does not have enough strength yet? This is difficult.
In a visual form, this can be shown as follows: if you take two squares of suitable sizes and disassemble into unit squares, then from this heap of unit squares you get the third square (Fig. 2): 
And if we do the same with the third dimension (Fig. 3), it will not work. Not enough cubes, or extra ones remain:
But the mathematician of the 17th century, Frenchman Pierre de Fermat, enthusiastically investigated the general equation x n + y n \u003d z n. And finally, I came to the conclusion: for n\u003e 2, there are no integer solutions. Fermat's proof is irretrievably lost. The 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 had a reputation for never making mistakes. Even if he did not leave evidence of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n \u003d 4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.
After Fermat, such great minds as Leonard Euler worked on the search for proof (in 1770 he proposed a solution for n \u003d 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n \u003d 5 in 1825), Gabriel Lame (who found a proof for n \u003d 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 saga of searching for a proof of Fermat's last theorem was practically over.
It is easy to show that it suffices to prove Fermat's theorem only for prime n: 3, 5, 7, 11, 13, 17, ... For composite n, the proof remains valid. But there are also infinitely many primes ...
In 1825, applying the method of Sophie Germain, women mathematicians, Dirichlet and Legendre independently proved the theorem for n \u003d 5. In 1839, using the same method, the Frenchman Gabriel Lame showed the truth of the theorem for n \u003d 7. Gradually the theorem was proved for almost all n less than one hundred.
Finally, the German mathematician Ernst Kummer showed in a brilliant study that the theorem in general form cannot be proved by 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, was not awarded.
In 1907, the wealthy German industrialist Paul Wolfskel, out of unrequited love, decided to take his own life. As a true German, he set the date and time for the suicide: exactly at midnight. On the last day, he drew up a will and wrote letters to friends and relatives. Business was over before midnight. I must say that Paul was interested in mathematics. Out of nothing else to do, he went to the library and began to read the famous article by Kummer. Suddenly it seemed to him that Kummer made a mistake in the course of his reasoning. Wolfskel began to parse this passage of the article, pencil in hand. Midnight has passed, morning has come. The gap in the evidence was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.
He soon died a natural death. The heirs were quite surprised: 100,000 marks (over 1,000,000 of the 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 due to the prover of Fermat's theorem. Not a pfennig was supposed to refute the theorem ...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and firmly refused to waste time on such a useless exercise. But the amateurs frolicked wonderfully. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E.M. Landau, whose duty was to analyze the submitted evidence, handed out cards to his students:
Dear. ... ... ... ... ... ... ...
Thank you for the manuscript you sent me with the proof of Fermat's Last Theorem. The first error is on page ... in line .... Because of her, all evidence is invalidated.
Professor E. M. Landau
In 1963, Paul Cohen, relying on Gödel's conclusions, proved the undecidability of one of Hilbert's twenty-three problems - the continuum hypothesis. But what if Fermat's Last Theorem is also undecidable ?! But true fanatics of the Great Theorem were not disappointed in the least. The advent of computers suddenly gave mathematicians a new method of proof. After World War II, groups of programmers and mathematicians proved Fermat's Last Theorem for all values \u200b\u200bof 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 claimed that Fermat's Last Theorem was true for all values \u200b\u200bof n to 4 million. But if we 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 rows of numbers, each with its own row. By chance, Taniyama compared these series with the series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. Connections have never been found between such different objects.
Nevertheless, friends, after careful testing, put forward a hypothesis: each elliptic equation has a double - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole direction in mathematics, but until the Taniyama-Shimura hypothesis was proved, 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 cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama-Shimura conjecture. Having proved that any elliptic curve is modular, we conclude that an elliptic equation with a solution to Fermat's equation does not exist, and Fermat's Last Theorem would be proved immediately. But for thirty years, it was not possible to prove the Taniyama-Shimura hypothesis, and there were less and less hopes for success.
In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. A schoolboy, student, graduate student, he prepared himself for this task.
Upon learning of Ken Ribet's findings, Wiles went headlong into proving the Taniyama-Shimura hypothesis. He decided to work in complete isolation and secrecy. "I understood that everything that has something to do with Fermat's Last Theorem is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal." Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.
In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), On which work lasted more than seven years.
While the hype in the press continued, 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 hectic summer waiting for the reviewers' feedback, hoping he could get their approval. At the end of August, experts found an insufficiently substantiated judgment.
It turned out that this solution contains a gross error, although it is generally correct. Wiles did not give up, called for the help of a well-known expert in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took as many as 130 (!) Pages in the mathematical journal "Annals of Mathematics". But the story did not end there either - the last point was only put 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 Nadia with the manuscript of the complete proof” (Andrew Waltz). Have I said that mathematicians are strange people?
This time, there was no doubt about the proof. 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 about the undecidability of Ferm's Last Theorem. But even those who know about the found proof continue to work in this direction - very few people are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the forces of very 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 ...
source
1Ivliev Yu.A.
The article is devoted to the description of a fundamental mathematical error made in the process of proving Fermat's Last Theorem at the end of the 20th century. The detected error not only distorts the true meaning of the theorem, but also prevents the development of a new axiomatic approach to the study of the powers of numbers and the natural series of numbers.
In 1995, an article was published, similar in size to a book, and reported on the proof of the famous Great (Last) Fermat's theorem (WTF) (for the history of the theorem and attempts to prove it, see, for example,). After this event, many scientific articles and popular science books appeared, promoting this proof, however, none of these works revealed a fundamental mathematical error in it, which crept in not even through the fault of the author, but through some strange optimism that gripped the minds mathematicians who dealt with this problem and related issues. The psychological aspects of this phenomenon were investigated in. It also provides a detailed analysis of the oversight that occurred, which is not of a particular nature, but is a consequence of a misunderstanding of the properties of the powers of integers. As shown in, Fermat's problem is rooted in a new axiomatic approach to the study of these properties, which has not yet been applied in modern science. But he got in the way of an erroneous proof, which gave the specialists in the theory of numbers false reference points and led researchers of Fermat's problem away from its direct and adequate solution. This work is devoted to removing this obstacle.
1. Anatomy of a mistake made in the course of proving the WTF
In the course of very long and tedious reasoning, Fermat's original assertion was reformulated in terms of a comparison of the pth degree Diophantine equation with elliptic curves of the third order (see Theorems 0.4 and 0.5 c). Such a comparison forced the authors of the actually collective proof to declare that their method and reasoning lead to the final solution of Fermat's problem (recall that the WTF had no recognized evidence for the case of arbitrary integer powers of integers until the 90s of the last century). The purpose of this consideration is to establish the mathematical incorrectness of the above comparison and, as a result of the analysis, to find a fundamental error in the proof presented in Art.
a) Where and what is the error?
So, let's go through the text, where on p. 448 it is said that after G. Frey's “witty idea” the possibility of proving the WTF opened up. In 1984 G. Frey suggested and
K. Ribet later proved that the supposed elliptic curve, representing the hypothetical integer solution of Fermat's equation,
y 2 \u003d x (x + u p) (x - v p) (1)
cannot be modular. However, A. Wiles and R. Taylor proved that every semistable elliptic curve defined over the field of rational numbers is modular. This led to the conclusion about the impossibility of integer solutions of Fermat's equation and, consequently, the validity of Fermat's assertion, which, in Wiles's notation, was written as Theorem 0.5: let there be an equality
u p + v p + w p \u003d 0 (2)
where u, v, w - rational numbers, integer exponent p ≥ 3; then (2) is satisfied only if uvw = 0 .
Now, apparently, one should go back and critically comprehend why curve (1) was a priori perceived as elliptical and what is its real connection with Fermat's equation. Anticipating this question, A. Wiles refers to the work of Y. Hellegouarch, in which he found a way to match Fermat's equation (supposedly solvable in integers) with a hypothetical curve of the 3rd order. Unlike H. Frey, I. Elleguarsh did not associate his curve with modular forms, but his method of obtaining equation (1) was used to further advance the proof of A. Wiles.
Let's dwell on work. The author carries out his reasoning in terms of projective geometry. Simplifying some of its notation and bringing them in accordance with, we find that the Abelian curve
Y 2 \u003d X (X - β p) (X + γ p) (3)
the Diophantine equation
x p + y p + z p \u003d 0 (4)
where x, y, z are unknown integers, p is an integer exponent from (2), and the solutions of the Diophantine equation (4) α p, β p, γ p are used to write the Abelian curve (3).
Now, to make sure that this is an elliptic curve of the 3rd order, it is necessary to consider the variables X and Y in (3) on the Euclidean plane. To do this, we use the well-known rule of arithmetic for elliptic curves: if there are two rational points on a cubic algebraic curve and the line passing through these points intersects this curve at one more point, then the latter is also a rational point. The hypothetical equation (4) formally represents the law of addition of points on a straight line. If we change variables x p \u003d A, y p \u003d B, z p \u003d C and direct the line thus obtained along the X-axis in (3), then it will intersect the curve of the 3rd degree at three points: (X \u003d 0, Y \u003d 0), (X \u003d β p, Y \u003d 0), (X \u003d - γ p, Y \u003d 0), which is reflected in the notation of the Abelian curve (3) and in a similar notation (1). However, is curve (3) or (1) actually elliptical? Obviously not, because the segments of the Euclidean line when adding points on it are taken on a nonlinear scale.
Returning to the linear coordinate systems of Euclidean space, instead of (1) and (3) we obtain formulas that are quite different from the formulas for elliptic curves. For example, (1) could be the following form:
η 2p \u003d ξ p (ξ p + u p) (ξ p - v p) (5)
where ξ p \u003d x, η p \u003d y, and the appeal to (1) in this case for the derivation of the WTF seems to be illegal. Despite the fact that (1) satisfies some criteria for the class of elliptic curves, it does not satisfy the most important criterion of being a third-degree equation in a linear coordinate system.
b) Error classification
So, once again, let's return to the beginning of the consideration and trace how it is drawn to the conclusion about the truth of the WTF. First, it is assumed that there is a solution to Fermat's equation in positive integers. Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (plane curve of degree 3) under the assumption that the elliptic curves obtained in this way exist (the second unconfirmed assumption). Thirdly, since it is proved by other methods that the constructed concrete curve is non-modular, it means that it does not exist. Hence the conclusion follows: there is no integer solution of Fermat's equation and, therefore, the WTF is correct.
There is one weak link in this reasoning, which after a detailed check turns out to be an error. This error occurs at the second stage of the proof process, when it is assumed that the hypothetical solution of Fermat's equation is at the same time a solution to a third-degree algebraic equation describing an elliptic curve of a known form. In itself, such an assumption would be justified if the indicated curve were indeed elliptical. However, as seen from item 1a), this curve is presented in nonlinear coordinates, which makes it "illusory", i.e. does not really exist in a linear topological space.
Now we need to clearly classify the found error. It consists in the fact that what needs to be proved is given as an argument of the proof. In classical logic, this error is known as a “vicious circle”. In this case, the integer solution of the Fermat equation is compared (apparently, presumably unambiguously) with a fictitious, non-existent elliptic curve, and then all the pathos of further reasoning goes to prove that a specific elliptic curve of this form, obtained from hypothetical solutions of the Fermat equation, does not exist.
How did it happen that such an elementary mistake was overlooked in serious mathematical work? Probably, this happened due to the fact that earlier in mathematics "illusory" geometric figures of the indicated type were not studied. Indeed, who could be interested, for example, in a fictitious circle obtained from Fermat's equation by changing the variables x n / 2 \u003d A, y n / 2 \u003d B, z n / 2 \u003d C? After all, its equation C 2 \u003d A 2 + B 2 does not have integer solutions for integers x, y, z and n ≥ 3. In the nonlinear coordinate axes X and Y, such a circle would be described by an equation that looks very similar to the standard shape:
Y 2 \u003d - (X - A) (X + B),
where A and B are no longer variables, but concrete numbers determined by the above replacement. But if the numbers A and B are given their original form, which consists in their exponential nature, then the inhomogeneity of the designations in the factors on the right side of the equation immediately catches the eye. This feature helps to distinguish illusion from reality and move from nonlinear coordinates to linear ones. On the other hand, if we consider numbers as operators when comparing them with variables, as, for example, in (1), then both should be homogeneous quantities, i.e. must have the same degrees.
This understanding of the powers of numbers as operators also allows us to see that the comparison of Fermat's equation with an illusory elliptic curve is not unambiguous. Take, for example, one of the factors on the right-hand side of (5) and expand it into p linear factors, introducing a complex number r such that r p \u003d 1 (see for example):
ξ p + u p \u003d (ξ + u) (ξ + r u) (ξ + r 2 u) ... (ξ + r p-1 u) (6)
Then form (5) can be represented as a decomposition into prime factors of complex numbers similar to the algebraic identity (6); however, the uniqueness of such an decomposition in the general case is questionable, which was shown by Kummer at one time.
2. Conclusions
It follows from the previous analysis that the so-called arithmetic of elliptic curves is not capable of shedding light on where to look for a proof of the WTF. After work, Fermat's statement, by the way, taken as an epigraph to this article, began to be perceived as a historical joke or a practical joke. However, in fact, it turns out that it was not Fermat who joked, but the specialists who gathered at the mathematical symposium in Oberwolfach in Germany in 1984, at which Frei voiced his witty idea. The consequences of such an imprudent statement brought mathematics as a whole to the brink of losing public confidence, which is described in detail in and which inevitably raises the question of the responsibility of scientific institutions to society before science. Comparison of Fermat's equation with Frey's curve (1) is the "lock" of the whole proof of Wiles concerning Fermat's theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof either.
Recently, various Internet reports have appeared that as if some prominent mathematicians have finally figured out Wiles's proof of Fermat's theorem, having come up with an excuse for it in the form of a "minimal" recalculation of integer points in Euclidean space. However, no innovations can cancel the classical results already obtained by humanity in mathematics, in particular, the fact that although any ordinal number coincides with its quantitative analogue, it cannot be a substitute for it in the operations of comparing numbers among themselves, and hence with inevitably it follows that the Frey curve (1) is not elliptic initially, i.e. is not by definition.
LIST OF REFERENCES:
- Ivliev Yu.A. Reconstruction of the native proof of Fermat's Last Theorem - Joint Scientific Journal (section "Mathematics"). April 2006 № 7 (167) p. 3-9, see also Pratsi Lugansk viddilennya International Academy of informatization. Ministry of Education of Science of Ukraine. Skhidnoukranskiy National University im. V. Dahl. 2006 No. 2 (13) p.19-25.
- Ivliev Yu.A. The greatest scientific scam of the twentieth century: the "proof" of Fermat's last theorem - Natural and technical sciences (section "History and methodology of mathematics"). August 2007 No. 4 (30) p.34-48.
- Edwards H.M. Fermat's last theorem. Genetic introduction to algebraic number theory. Per. from English. ed. B.F.Skubenko. M .: Mir 1980, 484 p.
- Hellegouarch Y. Points d´ordre 2p h sur les courbes elliptiques - Acta Arithmetica. 1975 XXVI p. 253-263.
- Wiles A. Modular elliptic curves and Fermat´s Last Theorem - Annals of Mathematics. May 1995 v. 141 Second series No. 3 p.443-551.
Bibliographic reference
Ivliev Yu.A. WYLES 'ERROR PROOF OF THE GREAT FARM'S THEOREM // Fundamental Research. - 2008. - No. 3. - S. 13-16;URL: http://fundamental-research.ru/ru/article/view?id\u003d2763 (date of access: 09/25/2019). We bring to your attention the journals published by the "Academy of Natural Sciences"
Fermat's Last Theorem Singh Simon
"Is Fermat's Last Theorem Proven?"
It was only the first step towards proving the Taniyama-Shimura conjecture, but Wiles's strategy was a brilliant mathematical breakthrough, a result worthy of publication. But due to the vow of silence, imposed by Wiles on himself, he could not tell the rest of the world about the result and had no idea who else could make such a significant breakthrough.
Wiles recalls his philosophical attitude towards any potential rival: “Nobody wants to spend years proving something and find that someone else has managed to find proof a few weeks earlier. But, oddly enough, since I was trying to solve a problem that was essentially considered insoluble, I was not very afraid of competitors. I just didn't hope that I or anyone else would come up with an idea that would lead to a proof. "
On March 8, 1988, Wiles was shocked to see headlines in large print on the front pages of newspapers that read "Fermat's Last Theorem Proven." The Washington Post and New York Times reported that thirty-eight-year-old Ioichi Miyaoka of Tokyo Metropolitan University had solved the world's most difficult mathematical problem. Miyaoka has not yet published his proof, but outlined its progress in a seminar at the Max Planck Institute for Mathematics in Bonn. Don Tsagir, who was present at Miyaoka's talk, expressed the optimism of the mathematical community in the following words: “The proof presented by Miyaoka is extremely interesting, and some mathematicians believe that it is highly likely to be correct. There is no certainty yet, but so far the evidence looks very encouraging. "
Speaking at a seminar in Bonn, Miyaoka spoke about his approach to solving the problem, which he considered from a completely different, algebraic-geometric point of view. Over the past decades, geometers have achieved a deep and subtle understanding of mathematical objects, in particular, the properties of surfaces. In the 70s the Russian mathematician S. Arakelov tried to establish parallels between the problems of algebraic geometry and the problems of number theory. This was one of the directions of Langlands' program, and mathematicians hoped that the unsolved problems of number theory could be solved by studying the corresponding problems of geometry, which also remained unsolved. Such a program was known as the philosophy of concurrency. Those algebraic geometers who tried to solve problems in number theory were called "arithmetic algebraic geometers". In 1983, they heralded their first significant victory when Gerd Faltings of the Princeton Institute for Advanced Study made a significant contribution to the understanding of Fermat's Theorem. We recall that, according to Fermat, the equation
at n the largest 2 have no integer solutions. Faltings decided that he was able to make progress in proving Fermat's Last Theorem by studying geometric surfaces associated with various values n... Surfaces associated with Fermat's equations at different values n, differ from each other, but have one common property - they all have through holes, or, more simply, holes. These surfaces are four-dimensional, as are the graphs of modular shapes. Two-dimensional sections of two surfaces are shown in Fig. 23. The surfaces associated with Fermat's equation look similar. The greater the value n in the equation, the more holes in the corresponding surface.
Fig. 23. These two surfaces were obtained using the computer program "Mathematica". Each of them represents a locus of points satisfying the equation x n + y n = z n (for the surface on the left n\u003d 3, for the surface on the right n\u003d 5). Variables x and y are considered complex here
Faltings succeeded in proving that, since such surfaces always have several holes, the associated Fermat equation could have only a finite set of integer solutions. The number of solutions could be anything - from zero, as Fermat assumed, to a million or a billion. Thus, Faltings did not prove Fermat's Last Theorem, but at least was able to reject the possibility of the existence of infinitely many solutions for Fermat's equation.
Five years later, Miyaoka reported that he had managed to move one step further. He was then in his twenties. Miyaoka formulated a conjecture about some inequality. It became clear that proving his geometric conjecture would mean proving that the number of solutions to Fermat's equation is not just finite, but equal to zero. Miyaoka's approach was similar to Wiles's in that they both tried to prove Fermat's Last Theorem by linking it to a fundamental conjecture in another area of \u200b\u200bmathematics. For Miyaoka it was algebraic geometry; for Wiles, the way to the proof lay through elliptic curves and modular forms. Much to Wiles's chagrin, he was still struggling to prove the Taniyama-Shimura conjecture when Miyaoka claimed that he had a complete proof of his own conjecture, and therefore Fermat's Last Theorem.
Two weeks after his Bonn talk, Miyaoka published five pages of calculations that formed the core of his proof, and a thorough examination began. Specialists in number theory and algebraic geometry all over the world studied, line by line, published calculations. A few days later, mathematicians discovered a contradiction in the proof that could not but cause concern. One part of Miyaoki's work led to a statement from number theory, from which, when translated into the language of algebraic geometry, a statement was obtained that contradicted a result obtained several years earlier. Although this did not necessarily invalidate all of Miyaoka's proof, the revealed contradiction did not fit into the philosophy of parallelism between number theory and geometry.
Two weeks later, Gerd Faltings, who paved the way for Miyaoka, announced that he had discovered the exact cause of the apparent concurrency violation - a gap in reasoning. The Japanese mathematician was a geometer and was not absolutely strict in translating his ideas into less familiar territory of number theory. An army of number theorists made desperate efforts to patch the hole in Miyaoka's proof, but to no avail. Two months after Miyaoka claimed to have a complete proof of Fermat's Last Theorem, the mathematical community came to a unanimous conclusion that Miyaoka's proof was doomed to fail.
As with previous failed proofs, Miyaoka managed to obtain many interesting results. Some fragments of his proof deserved attention as very ingenious applications of geometry to number theory, and in subsequent years other mathematicians used them to prove some theorems, but no one was able to prove Fermat's Last Theorem in this way.
The hype about Fermat's Last Theorem soon died down, and newspapers published brief notes stating that the three-hundred-year-old conundrum remained unsolved. The following inscription appeared on the wall of the New York subway station on Eighth Street, undoubtedly inspired by the publications in the press about Fermat's Last Theorem: “The equation xn + yn = zn has no solutions. I have found a truly amazing proof of this fact, but I cannot write it here, because my train came. "
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