In the 17th century, a lawyer and part-time mathematician Pierre Fermat lived in France, who devoted long hours of leisure to his hobby. One winter evening, sitting by the fireplace, he put forward one most curious statement from the field of number theory - it was this which was later called the Great or Great Theorem of Fermat. Perhaps the hype would not have been so significant in mathematical circles if one event had not happened. The mathematician often spent evenings studying the favorite book of Diophantus of Alexandria "Arithmetic" (3rd century), while writing down important thoughts in its margins - this rarity was carefully preserved for posterity by his son. So, on the wide margins of this book, Fermat's hand left the following inscription: "I have a rather startling proof, but it is too large to fit in the margins." It was this entry that caused the overwhelming excitement around the theorem. The mathematicians did not doubt that the great scientist declared that he had proved his own theorem. You are probably wondering: "Did he actually prove it, or it was a banal lie, or maybe there are other versions of why this entry, which did not let mathematicians of subsequent generations sleep peacefully, ended up in the margins of the book?"
The essence of the Great Theorem
The fairly well-known Fermat's theorem is simple in its essence and consists in the fact that provided that n is greater than two, a positive number, the equation X n + Y n \u003d Z n will not have solutions of type zero within the framework natural numbers... In this seemingly simple formula, incredible complexity was masked, and it was fought for three centuries to prove it. There is one oddity - the theorem was late with the birth of the world, since its special case for n \u003d 2 appeared 2200 years ago - this is the no less famous Pythagorean theorem.
It should be noted that the history of the well-known Fermat's theorem is very instructive and entertaining, and not only for mathematicians. What is most interesting is that science was not a job for a scientist, but a simple hobby, which in turn gave the Farmer great pleasure. He also constantly kept in touch with a mathematician, and also a friend, shared ideas, but oddly enough, he did not seek to publish his own works.
The works of the mathematician Farmer
As for the works of Farmer themselves, they were found precisely in the form of ordinary letters. In some places there were no whole pages, and only scraps of correspondence survived. More interesting is the fact that for three centuries scientists have been looking for the theorem that was discovered in the works of Farmer.
But whoever dared to prove it, attempts were reduced to "zero". The famous mathematician Descartes even accused the scientist of boasting, but it all boiled down to just the most common envy. In addition to the creation, the Farmer also proved his own theorem. True, the solution was found for the case where n \u003d 4. As for the case for n \u003d 3, it was revealed by the mathematician Euler.
How they tried to prove Farmer's theorem
At the very beginning of the 19th century, this theorem continued to exist. Mathematicians found many theorem proofs that limited themselves to natural numbers within two hundred.
And in 1909, a rather large sum was put on the line, equal to one hundred thousand marks of German origin - and all this was just to solve the problem associated with this theorem. The prize fund itself was left by a wealthy mathematics lover Paul Wolfskel, a native of Germany, by the way, it was he who wanted to "lay hands on himself", but thanks to such involvement in the Farmer's theorem, he wanted to live. The resulting excitement generated tons of "proofs" that flooded German universities, and in the circle of mathematicians the nickname "Fermist" was born, which was half-contemptuously called any ambitious upstart who could not provide clear evidence.
Hypothesis of Japanese mathematician Yutaka Taniyama
There were no shifts in the history of the Great Theorem until the middle of the 20th century, but one interesting event did take place. In 1955, a mathematician from Japan, Yutaka Taniyama, who was 28 years old, showed the world a statement from a completely different mathematical field - his hypothesis, unlike Fermat, was ahead of its time. It reads: "Each elliptic curve corresponds to a certain modular shape." It seems to be absurd for every mathematician, like that wood is made of a certain metal! The paradoxical hypothesis, like most other stunning and brilliant discoveries, was not accepted, since they simply had not yet matured to it. And Yutaka Taniyama committed suicide, three years later - an inexplicable act, but, probably, honor for a true samurai genius was above all.
For a whole decade, the hypothesis was not remembered, but in the seventies it rose to the peak of popularity - it was confirmed by everyone who could understand it, but, like Fermat's theorem, it remained unproven.
How Taniyama's conjecture and Fermat's theorem are related
15 years later, a key event occurred in mathematics, and it combined the hypothesis of the famous Japanese and Fermat's theorem. Gerhard Gray stated that when Taniyama's conjecture is proved, then there will be a proof of Fermat's theorem. That is, the latter is a consequence of Taniyama's hypothesis, and after a year and a half, Fermat's theorem was proved by a professor at the University of California, Kenneth Ribet.
As time went on, regression was replaced by progress, and science was rapidly advancing, especially in the field of computer technology. Thus, the value of n began to increase more and more.
At the very end of the 20th century, the most powerful computers were located in military laboratories, programming was carried out to derive a solution to the problem of the well-known Fermat. As a consequence of all attempts, it was found that this theorem is correct for many values \u200b\u200bof n, x, y. But, unfortunately, this did not become the final proof, since there was no specifics as such.
John Wiles proved the great Fermat's Theorem
And finally, only at the end of 1994, a mathematician from England, John Wiles, found and demonstrated an exact proof of the controversial Farmer's theorem. Then, after many improvements, discussions on this matter came to their logical conclusion.
The rebuttal was posted on over a hundred pages of one magazine! Moreover, the theorem was proved on a more modern apparatus higher mathematics... And surprisingly, at the time when the Farmer wrote his work, such an apparatus did not exist in nature. In a word, the person was recognized as a genius in this area, with which no one could argue. Despite everything that happened, today one can be sure that the presented theorem of the great scientist Farmer is justified and proven, and no mathematician with common sense will start a dispute on this topic, with which even the most inveterate skeptics of all mankind agree.
The full name of the person after whom the presented theorem was named was Pierre de Fermer. He has contributed to a wide variety of areas of mathematics. But, unfortunately, most of his works were published only after his death.
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 integersThe proof of the tautology is obviously wrong 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 assumption made, 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 great 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).
On the basis of 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 cannot be a single 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 equation (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 members of the family (10) and (11) is equal to half of the product of 13 by 12 and 21 by 20, i.e. 78 and 210.
Each member of the 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 triplet 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 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 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 to 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 with 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, by 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 of the opposite theorem, the opposite of 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. 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 absurdity 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 .: ill.-C.112 /.
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 converse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If 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 the relation of 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.
Suppose that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. Let us formulate it in general form in short form So: of AND should E ... Symbol AND the given condition of the theorem, taken 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 .
As a result, we got 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 explanatory dictionary 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 resolution 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 a 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, and the other in this case is true. Both parts of the condition have the same origin, are accepted as data, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature was 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: only the converse theorem is proved by a logical method by contradiction, which has a direct theorem proved by a mathematical method and which cannot be proved by a mathematical method.
The resulting 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 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 different 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 Phenomenon of Wiles' Proofs")
In other words, Gerhard Frey assumed that the equation of the great Fermat 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 conservative 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.
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 proved by the usual mathematical method. In this case, it could be proved by contradiction. And since it is a direct theorem, 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 the 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, a number theorist, made the following authoritative statement titled: “ Brief retelling 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 miraculous 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. In this paragraph, 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 family of triplets (78 and 210 pieces) the BTF is fulfilled (and only for it) is devoid of 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 it should have suggested another method, which it 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 equation (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.
THE HISTORY OF THE GREAT FARM'S THEOREMA grand affair
Once in the New Year's issue of the mailing list on how to make toasts, I casually mentioned that at the end of the twentieth century there was one grand event that many did not notice - the so-called Fermat's Last Theorem was finally proved. On this occasion, among the letters I received, I found two responses from girls (one of them, as far as I remember, is a ninth-grader Vika from Zelenograd), who were surprised by this fact.
And I was surprised at how vividly the girls are interested in the problems of modern mathematics. Therefore, I think that not only girls, but also boys of all ages - from high school students to pensioners, will also be interested in learning the history of the Great Theorem.
The proof of Fermat's theorem is a great event. And since it is not customary to joke with the word "great", then it seems to me that every self-respecting orator (and all of us, when we say orators) are simply obliged to know the history of the theorem.
If it so happens that you do not like mathematics as I love it, then look through some deeper details with a cursory glance. Realizing that not all readers of our mailing list are interested in wandering in the mathematical wilderness, I tried not to give any formulas (except for the equation of Fermat's theorem and a couple of hypotheses) and to simplify as much as possible some specific issues.
How Ferma made porridge
The French lawyer and also the great mathematician of the 17th century Pierre Fermat (1601-1665) put forward one curious statement from the field of number theory, which later became known as the Great (or Great) Fermat's theorem. This is one of the most famous and phenomenal mathematical theorems. Probably, the excitement around her would not be so strong if in the book of Diophantus of Alexandria (III century AD) "Arithmetic", which Fermat often studied, making notes on its wide fields, and which was kindly preserved for posterity by his son Samuel , the following record of the great mathematician was not discovered:
"I have a very startling piece of evidence, but it is too large to fit in the margin."
It was it, this record, that was the reason for the subsequent grandiose turmoil around the theorem.
So, the famous scientist declared that he had proved his theorem. Let's ask ourselves a question: did he really prove it or did he trite lied? Or are there other versions explaining the appearance of that note in the margins, which did not allow many mathematicians of the next generations to sleep peacefully?
The story of the Great Theorem is as fascinating as an adventure in time. In 1636 Fermat stated that an equation of the form x n + y n \u003d z n has no integer solutions for exponent n\u003e 2. This is actually Fermat's Last Theorem. In this seemingly simple mathematical formula, the universe disguises incredible complexity. An American mathematician of Scottish descent, Eric Temple Bell, in his book The Last Problem (1961) even suggested that perhaps humanity will cease to exist before it can prove Fermat's Last Theorem.
It is somewhat strange that, for some reason, the theorem was late with its appearance, since the situation was ripe for a long time, because its special case for n \u003d 2 - another famous mathematical formula - the Pythagorean theorem, appeared twenty-two centuries earlier. Unlike Fermat's theorem, the Pythagorean theorem has an infinite set of integer solutions, for example, such Pythagorean triangles: (3,4,5), (5,12,13), (7,24,25), (8,15,17 ) ... (27,36,45) ... (112,384,400) ... (4232, 7935, 8993) ...
The Great Theorem Syndrome
Who just tried to prove Fermat's theorem. Any fledgling student considered it his duty to apply the Great Theorem, but no one was able to prove it. At first it did not work for a hundred years. Then another hundred. And further. A massive syndrome began to develop among mathematicians: "How is that? Fermat has proved, but I can’t, or what?" - and some of them have gone crazy on this basis in the full sense of the word.
No matter how much the theorem is checked, it always turns out to be true. I knew an energetic programmer who was obsessed with the idea of \u200b\u200bdisproving the Great Theorem, trying to find at least one solution (counterexample) of it by iterating over integers using a high-speed computer (at that time more often called a computer). He believed in the success of his enterprise and loved to say: "A little more - and a sensation will break out!" I think that in different parts of our planet there were a considerable number of such kind of daring seekers. Of course, he did not find a single solution. And no computers, even with fabulous speed, could ever verify the theorem, because all the variables of this equation (including the exponents) can grow to infinity.
Theorem requires proof
Mathematicians know that if a theorem is not proven, anything can follow from it (both true and false), as was the case with some other hypotheses. For example, in one of his letters, Pierre Fermat suggested that numbers of the form 2 n +1 (the so-called Fermat numbers) are necessarily prime (i.e., they do not have integer divisors and are divisible without remainder only by themselves and by one), if n is a power of two (1, 2, 4, 8, 16, 32, 64, etc.). This hypothesis of Fermat lived for more than a hundred years - until, in 1732, Leonard Euler showed that
2 32 +1 \u003d 4 294 967 297 \u003d 6 700 417 641
Then, almost 150 years later (1880) Fortunet Landry factorized the following Fermat's number:
2 64 +1 \u003d 18 446 744 073 709 551 617 \u003d 274 177 67 280 421 310 721
How they, without the help of computers, were able to find the divisors of these large numbers - God only knows. In turn, Euler put forward a hypothesis that the equation x 4 + y 4 + z 4 \u003d u 4 has no solution in integers. However, after about 250 years, in 1988, Naum Elkis from Harvard managed to discover (already with the help of a computer program) that
2 682 440 4 + 15 365 639 4 + 18 796 760 4 = 20 615 673 4
Therefore, Fermat's Last Theorem required a proof, otherwise it was just a hypothesis, and it could well be that somewhere out there in the endless number fields the solution of the Great Theorem equation was lost.
The most virtuoso and fruitful mathematician of the 18th century, Leonard Euler, whose records mankind has been raking for almost a century, proved Fermat's theorem for degrees 3 and 4 (or rather, he repeated the lost proofs of Pierre Fermat himself); his follower in number theory, Legendre (and also independently of him Dirichlet) - for degree 5; Lamé - for degree 7. But in general the theorem remained unproven.
March 1, 1847 at a meeting Paris Academy sciences, two outstanding mathematicians - Gabriel Lame and Augustin Cauchy - announced that they had come to the end of the proof of the Great Theorem and started a race, publishing their proofs in parts. However, the duel between them was interrupted because the same error was found in their proofs, which was pointed out by the German mathematician Ernst Kummer.
At the beginning of the 20th century (1908), a wealthy German entrepreneur, philanthropist and scientist Paul Wolfskel bequeathed one hundred thousand marks to anyone who would present a complete proof of Fermat's theorem. Already in the first year after the publication of Wolfskel's will by the Göttingent Academy of Sciences, it was inundated with thousands of proofs from amateurs of mathematics, and this stream did not stop for decades, but all of them, as you might guess, contained errors. They say that the academy prepared forms with approximately the following content:
Dear __________________________!
In your proof of Fermat's theorem on ____ page, ____ line above
the following error was encountered in the formula: __________________________ :,
Which were sent to unlucky prize seekers.
At that time, a half-disdainful nickname appeared in the circle of mathematicians - farmer... This was the name of any self-confident upstart who lacked knowledge, but more than enough ambition to hurry up to try his hand at proving the Great Theorem, and then, not noticing his own mistakes, proudly slapping his chest, loudly declare: “I was the first to prove Fermat's theorem! " Every farmer, even if he was even ten thousandth in a row, considered himself the first - and that was funny. The simple appearance of The Last Theorem reminded farmers of easy prey so much that they were absolutely not embarrassed that even Euler and Gauss could not cope with it.
(Ironists, oddly enough, still exist. One of them, although he did not think that he had proved the theorem, as a classical Fermist, but until recently made attempts - he refused to believe me when I told him that Fermat's theorem had already been proved).
The most powerful mathematicians, perhaps, in the quiet of their offices, also tried to carefully approach this overwhelming barbell, but they did not speak about it aloud, so as not to be known as farmers and, thus, not to harm their high authority.
By that time there was a proof of the theorem for the exponent n<100. Потом для n<619. Надо ли говорить о том, что все доказательства невероятно сложны. Но в общем виде теорема оставалась недоказанной.
Strange hypothesis
Until the middle of the twentieth century, no serious progress in the history of the Great Theorem was observed. But soon an interesting event took place in mathematical life. In 1955, 28-year-old Japanese mathematician Yutaka Taniyama put forward a statement from a completely different area of \u200b\u200bmathematics called the Taniyama hypothesis (aka the Taniyama-Shimura-Weil hypothesis), which, unlike Fermat's belated theorem, was ahead of its time.
Taniyama's hypothesis says: "Each elliptic curve corresponds to a certain modular shape." For mathematicians of that time, this statement sounded about as absurd as the statement sounds to us: "a certain metal corresponds to each tree." It is not difficult to guess how a normal person might react to such a statement - he simply will not take it seriously, which is what happened: mathematicians have unanimously ignored the hypothesis.
A little explanation. Elliptic curves, known for a long time, have a two-dimensional form (located on a plane). Modular functions, discovered in the 19th century, have a four-dimensional form, so we cannot even imagine them with our three-dimensional brains, but we can describe them mathematically; in addition, modular forms are amazing in that they have the utmost possible symmetry - they can be translated (shifted) in any direction, mirrored, interchanged fragments, rotated in infinitely many ways - without changing their appearance. As you can see, elliptic curves and modular forms have little in common. Taniyama's hypothesis asserts that the descriptive equations of these two absolutely different mathematical objects corresponding to each other can be expanded into one and the same mathematical series.
Taniyama's hypothesis was too paradoxical: it combined completely different concepts - rather simple flat curves and unimaginable four-dimensional shapes. It never occurred to anyone. When, at the International Mathematical Symposium in Tokyo in September 1955, Taniyama demonstrated several correspondences of elliptic curves to modular forms, everyone saw in this nothing more than amusing coincidences. To Taniyama's humble question: is it possible for each elliptic curve to find the corresponding modular function, the venerable Frenchman André Weil, who at that time was one of the world's best specialists in number theory, gave a completely diplomatic answer, which, they say, if the inquisitive Taniyama does not leave enthusiasm, then maybe he will be lucky, and his incredible hypothesis will be confirmed, but this must not happen soon. In general, like many other outstanding discoveries, at first Taniyama's hypothesis was ignored, because they had not yet matured to it - almost no one understood it. Only one colleague of Taniyama, Goro Shimura, knowing his highly gifted friend well, intuitively felt that his hypothesis was correct.
Three years later (1958) Yutaka Taniyama committed suicide (however, samurai traditions are strong in Japan). From the point of view of common sense, this is a completely incomprehensible act, especially when you consider that very soon he was going to get married. The leader of young Japanese mathematicians began his suicide note as follows: "Yesterday I did not think about suicide. Lately I often heard from others that I was tired mentally and physically. Actually, I still do not understand why I am doing this ..." and so on on three sheets. It's a pity, of course, that this is the fate of an interesting person, but all geniuses are a little strange - that's why they are geniuses (for some reason, the words of Arthur Schopenhauer came to mind: "in ordinary life, a genius is as good as a telescope in a theater") ... The hypothesis was orphaned. Nobody knew how to prove it.
For ten years Taniyama's hypothesis was hardly remembered. But in the early 70s it became popular - it was regularly checked by everyone who could figure it out - and it was always confirmed (like, in fact, Fermat's theorem), but, as before, no one could prove it.
The amazing connection between the two hypotheses
Another 15 years passed. In 1984, there was one key event in the life of mathematics, which combined the extravagant Japanese hypothesis with Fermat's Last Theorem. German Gerhard Frey put forward an interesting statement similar to the theorem: "If Taniyama's hypothesis is proved, then Fermat's Last Theorem will also be proved." In other words, Fermat's theorem is a consequence of Taniyama's conjecture. (Frey, using clever mathematical transformations, reduced Fermat's equation to the form of an elliptic curve equation (the same one that appears in Taniyama's hypothesis), more or less substantiated his assumption, but could not prove it). And just a year and a half later (1986), Professor of the University of California, Kenneth Ribet, clearly proved Frey's theorem.
What happened now? Now it turned out that, since Fermat's theorem is already precisely a consequence of Taniyama's hypothesis, it is only necessary to prove the latter in order to break the laurels of the conqueror of the legendary Fermat's theorem. But the hypothesis was not easy. In addition, mathematicians over the centuries developed an allergy to Fermat's theorem, and many of them decided that it would also be almost impossible to cope with Taniyama's hypothesis.
Fermat's death hypothesis. The birth of the theorem
Another 8 years have passed. One progressive English professor of mathematics from Princeton University (New Jersey, USA), Andrew Wiles, thought he had found a proof of Taniyama's conjecture. If the genius is not bald, then, as a rule, disheveled. Wiles is disheveled, hence looks like a genius. It was tempting to enter History, of course, and I really wanted to, but Wiles, as a real scientist, did not flatter himself, realizing that thousands of farmers before him also saw ghostly evidence. Therefore, before presenting his proof to the world, he carefully checked it himself, but realizing that he could have a subjective bias, he also attracted others to the checks, for example, under the guise of ordinary mathematical tasks, he sometimes threw various fragments of his proof to smart graduate students. Wiles later admitted that no one except his wife knew that he was working on a proof of the Great Theorem.
And now, after long checks and painful thoughts, Wiles finally plucked up the courage, or maybe, as it seemed to him, the impudence, and on June 23, 1993, at a mathematical conference on number theory in Cambridge, announced his great achievement.
This, of course, was a sensation. No one expected such agility from a little-known mathematician. The press immediately appeared. Everyone was tormented by burning interest. Slender formulas, like strokes of a beautiful picture, appeared before the curious eyes of the audience. Real mathematicians, after all, they are - they look at all sorts of equations and see in them not numbers, constants and variables, but hear music, like Mozart, looking at the stave. In the same way, when we read a book, we look at the letters, but it seems as if we do not notice them, but immediately perceive the meaning of the text.
The presentation of the proof seemed to be successful - no errors were found in it - no one heard a single false note (although most mathematicians just stared at it like first graders at the integral and did not understand anything). Everyone decided that a large-scale event had happened: Taniyama's hypothesis was proved, and therefore Fermat's Last Theorem. But about two months later, a few days before the manuscript of Wiles's proof was due to go to circulation, a discrepancy was found in it (Katz, a colleague of Wiles, noticed that one piece of reasoning relied on the "Euler system", but that built by Wiles, such a system was not), although in general Wiles' techniques were found to be interesting, elegant and innovative.
Wiles analyzed the situation and decided he had lost. One can imagine how he felt with all his being, which means "from the great to the ridiculous one step." "I wanted to enter History, but instead joined the team of clowns and comedians - arrogant farmers" - approximately such thoughts wore him out in that painful period of his life. For him, a serious scientist-mathematician, it was a tragedy, and he threw his proof on the back burner.
But a little over a year later, in September 1994, while thinking about that bottleneck of the proof with his colleague Taylor from Oxford, the latter suddenly dawned on the idea that the "Euler system" could be changed to Iwasawa's theory (section of number theory). Then they tried to use Iwasawa's theory, dispensing with the "Euler system", and they all came together. The corrected version of the proof was submitted for verification and a year later it was announced that everything in it was absolutely clear, without a single mistake. In the summer of 1995, one of the leading mathematical journals - Annals of Mathematics - published a complete proof of Taniyama's conjecture (hence, Fermat's Great (Great) Theorem), which took up the entire issue - over a hundred pages. The proof is so complex that only a few dozen people around the world could understand it in its entirety.
Thus, at the end of the twentieth century, the whole world recognized that in the 360 \u200b\u200byear of its life, Fermat's Last Theorem, which in fact had been a hypothesis all this time, became a proven theorem. Andrew Wiles proved the Great (Great) Fermat's Theorem and entered History.
Just think, they proved some kind of theorem ...
The happiness of the discoverer always goes to someone alone - it is he who, with the last blow of the hammer, breaks the hard nut of knowledge. But one cannot ignore the many previous blows, which for more than one century formed a crack in the Great Theorem: Euler and Gauss (the kings of mathematics of their times), Evariste Galois (who managed to found the theory of groups and fields in his short 21-year life, whose works were recognized as genius only after his death), Henri Poincaré (founder of not only bizarre modular forms, but also of conventionalism - a philosophical movement), David Gilbert (one of the strongest mathematicians of the 20th century), Yutaku Taniyama, Goro Shimuru, Mordell, Faltings, Ernst Kummer, Barry Mazur, Gerhard Frey, Ken Ribbet, Richard Taylor and others real scientists (I'm not afraid of these words).
The proof of Fermat's Last Theorem can be put on a par with such achievements of the twentieth century as the invention of the computer, the nuclear bomb and space flight. Although it is not so widely known, because it does not intrude into the zone of our momentary interests, such as a TV set or an electric light bulb, it was a supernova explosion that, like all immutable truths, will always shine on humanity.
You can say: "think about it, you proved some theorem, who needs it?". A fair question. Here David Gilbert's answer will do exactly. When, to the question:" What is the most important task for science now? ", He replied:" to catch a fly on the far side of the Moon ", he was reasonably asked:" a who needs it?", he replied:" Nobody needs it. But think about how many important, most difficult problems need to be solved to accomplish this. "Think how many problems in 360 years mankind was able to solve before proving Fermat's theorem. In search of its proof, almost half of modern mathematics was discovered. take into account that mathematics is the vanguard of science (and, by the way, the only science that is built without a single mistake), and any scientific achievements and inventions begin right here. As Leonardo da Vinci noted, “a science can only be recognized as a teaching that is mathematically confirmed ".
* * *
And now let's go back to the beginning of our story, recall Pierre Fermat's entry in the margins of the Diophantus textbook and once again ask ourselves the question: did Fermat really prove his theorem? This, of course, we cannot know for sure, and how in any business different versions arise here:
Version 1: Fermat proved his theorem. (When asked, "Did Fermat have exactly the same proof of his theorem?", Andrew Wiles remarked: "Fermat could not have so proof. This is proof of the twentieth century. "You and I understand that in the seventeenth century mathematics, of course, was not the same as at the end of the twentieth century - in that era, Artanyana, the queen of sciences did not yet possess those discoveries (modular forms, Taniyama's theorems , Freya, etc.), which only allowed to prove Fermat's Last Theorem. Of course, one can assume: what the hell is not joking - what if Fermat guessed in a different way? This version, although probable, but according to the estimates of most mathematicians, is practically impossible);
Version 2: Pierre Fermat thought that he had proved his theorem, but there were mistakes in his proof. (That is, Fermat himself was also the first farmer);
Version 3: Fermat did not prove his theorem, but in the fields he simply lied.
If one of the last two versions is correct, which is most likely, then a simple conclusion can be drawn: great people, although they are great, they can also be wrong or sometimes they are not averse to lying (basically, this conclusion will be useful for those who are inclined to completely trust their idols and other rulers of thoughts). Therefore, reading the works of the authoritative sons of humanity or listening to their pretentious speeches, you have every right to doubt their statements. (Please note that to doubt is not to reject).
The reprint of the article materials is possible only with the obligatory links to the site (on the Internet - hyperlink) and to the author
It is unlikely that at least one year in the life of our editorial office passed without receiving a dozen proofs of Fermat's theorem. Now, after the "victory" over her, the flow has subsided, but has not dried up.
Of course, not to dry it completely, we publish this article. And not to justify ourselves - that, they say, that is why we kept silent, we ourselves had not matured to discuss such complex problems.
But if the article really seems complicated, look right at the end of it. You will have to feel that passions have subsided temporarily, science is not over, and soon new proofs of new theorems will be sent to the editorial office.
It seems that the 20th century was not in vain. First, people created a second Sun for a moment by detonating a hydrogen bomb. Then they walked on the moon and finally proved the notorious Fermat's theorem. Of these three miracles, the first two are on everyone's lips, for they have caused huge social consequences. On the contrary, the third miracle looks like another scientific toy - on a par with the theory of relativity, quantum mechanics, and Gödel's theorem on the incompleteness of arithmetic. However, relativity and quanta led physicists to the hydrogen bomb, and the research of mathematicians filled our world with computers. Will this series of miracles continue into the 21st century? Is it possible to trace the connection between the next scientists toys and revolutions in our life? Does this connection allow for successful predictions? Let's try to understand this using the example of Fermat's theorem.
Let us first note that she was born much later than her natural term. After all, the first special case of Fermat's theorem is the Pythagorean equation X 2 + Y 2 \u003d Z 2, connecting the lengths of the sides of a right-angled triangle. Having proved this formula twenty-five centuries ago, Pythagoras immediately asked the question: are there many triangles in nature in which both legs and hypotenuse have an integer length? It seems that the Egyptians knew only one such triangle - with sides (3, 4, 5). But it is not hard to find other options: for example (5, 12, 13), (7, 24, 25) or (8, 15, 17). In all these cases, the length of the hypotenuse has the form (A 2 + B 2), where A and B are coprime numbers of different parity. In this case, the lengths of the legs are equal (A 2 - B 2) and 2AB.
Noticing these relations, Pythagoras easily proved that any triple of numbers (X \u003d A 2 - B 2, Y \u003d 2AB, Z \u003d A 2 + B2) is a solution to the equation X 2 + Y 2 \u003d Z 2 and defines a rectangle with mutually simple side lengths. It is also seen that the number of different triplets of this kind is infinite. But do all solutions of the Pythagorean equation have this form? Pythagoras could neither prove nor refute such a hypothesis and left this problem to his descendants, without focusing on it. Who wants to highlight their failures? It seems that after that the problem of integer right-angled triangles lay in oblivion for seven centuries - until a new mathematical genius named Diophantus appeared in Alexandria.
We know little about him, but it is clear: he was not at all like Pythagoras. He felt like a king in geometry and even beyond it - be it in music, astronomy or politics. The first arithmetic connection between the lengths of the sides of the euphonic harp, the first model of the Universe from concentric spheres carrying planets and stars, with the Earth in the center, finally, the first republic of scientists in the Italian city of Crotone - these are the personal achievements of Pythagoras. What could Diophantus, a modest researcher at the great Museum, which long ago ceased to be the pride of the city crowd, could oppose such successes?
Only one thing: a better understanding of the ancient world of numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet possess the positional system of writing large numbers, but he knew what negative numbers are and, probably, spent many hours thinking about why the product of two negative numbers is positive. The world of integers was first revealed to Diophantus as a special universe, different from the world of stars, segments or polyhedra. The main occupation of scientists in this world is solving equations, a real master finds all possible solutions and proves that there are no other solutions. This is what Diophantus did with the quadratic equation of Pythagoras, and then he wondered: does at least one solution have a similar cubic equation X 3 + Y 3 \u003d Z 3?
Diophantus failed to find such a solution; his attempt to prove that there were no solutions was also unsuccessful. Therefore, formalizing the results of his works in the book "Arithmetic" (this was the world's first textbook of number theory), Diophantus analyzed the Pythagorean equation in detail, but did not mention a word about possible generalizations of this equation. But he could: after all, it was Diophantus who first proposed the notation for the powers of integers! But alas: the concept of "problem book" was alien to Hellenic science and pedagogy, and it was considered indecent to publish lists of unsolved problems (only Socrates acted differently). If you cannot solve the problem - shut up! Diophantus fell silent, and this silence dragged on for fourteen centuries - until the onset of modern times, when interest in the process of human thought was revived.
Who just fantasized about what at the turn of the XVI - XVII centuries! The indefatigable calculator Kepler tried to guess the relationship between the distances from the Sun to the planets. Pythagoras did not succeed. Kepler was successful after learning how to integrate polynomials and other simple functions. On the contrary, the dreamer Descartes did not like long calculations, but it was he who first presented all points of a plane or space as sets of numbers. This daring model reduces any geometric figure problem to some algebraic equation problem - and vice versa. For example, entire solutions of the Pythagorean equation correspond to integer points on the surface of a cone. The surface corresponding to the cubic equation X 3 + Y 3 \u003d Z 3 looks more complicated, its geometric properties did not suggest anything to Pierre Fermat, and he had to make new paths through the jungle of integers.
In 1636, a young lawyer from Toulouse got hold of a book of Diophantus, just translated into Latin from a Greek original, which accidentally survived in some Byzantine archive and was brought to Italy by one of the Roman fugitives at the time of the Turkish ruin. Reading an elegant discourse on the Pythagorean equation, Fermat wondered: is it possible to find such a solution to it, which consists of three square numbers? There are no small numbers of this kind: it is easy to check by brute force. What about big decisions? Without a computer, Fermat could not carry out a numerical experiment. But he noticed that for each "large" solution of the equation X 4 + Y 4 \u003d Z 4 it is possible to construct a smaller solution. This means that the sum of the fourth powers of two integers is never equal to the same power of the third! What about the sum of two cubes?
Inspired by the success for degree 4, Fermat tried to modify the "descent method" for degree 3 - and he succeeded. It turned out that it is impossible to make two small cubes from those unit cubes, into which a large cube with a whole edge length fell apart. The Triumphant Fermat made a short note in the margin of the book of Diophantus and sent a letter to Paris detailing his discovery. But he did not receive an answer - although usually metropolitan mathematicians quickly reacted to the next success of their lone rival colleague in Toulouse. What's the matter here?
Quite simply: by the middle of the 17th century, arithmetic was out of fashion. The great successes of the Italian algebraists of the 16th century (when the equations-polynomials of degrees 3 and 4 were solved) did not become the beginning of a general scientific revolution, because they did not allow solving new bright problems in adjacent fields of science. Now, if Kepler managed to guess the orbits of the planets using pure arithmetic ... But alas, this required a mathematical analysis. This means that it must be developed - right up to the complete triumph of mathematical methods in natural science! But analysis grows out of geometry, while arithmetic remains a field of fun for idle lawyers and other lovers of the eternal science of numbers and figures.
Thus, Fermat's arithmetic successes were untimely and remained invaluable. He was not upset by this: for the glory of a mathematician, the facts of differential calculus, analytic geometry and probability theory, which were first discovered to him, were enough. All these discoveries by Fermat immediately entered the golden fund of the new European science, while the theory of numbers faded into the background for another hundred years - until it was revived by Euler.
This "king of mathematicians" of the 18th century was a champion in all applications of analysis, but he did not neglect arithmetic either, since new methods of analysis led to unexpected facts about numbers. Who would have thought that the infinite sum of inverse squares (1 + 1/4 + 1/9 + 1/16 +…) is equal to π 2/6? Who among the Hellenes could have foreseen that similar series would prove the irrationality of π?
Such successes forced Euler to carefully re-read the surviving manuscripts of Fermat (fortunately, the son of the great Frenchman managed to publish them). True, the proof of the "grand theorem" for degree 3 has not survived, but Euler easily restored it from just one indication of the "descent method", and immediately tried to transfer this method to the next prime degree - 5.
It was not so! Complex numbers appeared in Euler's reasoning, which Fermat managed to overlook (this is the usual lot of discoverers). But factoring complex integers is a delicate matter. Even Euler did not understand it to the end and put the "Fermat problem" aside, hurrying to complete his main work - the textbook "Fundamentals of Analysis", which was supposed to help every talented youth to get on a par with Leibniz and Euler. The publication of the textbook was completed in St. Petersburg in 1770. But Euler did not return to Fermat's theorem, being sure that everything that his hands and mind touched would not be forgotten by the new scientist youth.
And so it happened: the Frenchman Adrien Legendre became Euler's successor in number theory. At the end of the 18th century, he completed the proof of Fermat's theorem for degree 5 - and although it failed for large simple degrees, he compiled another textbook on number theory. Let his young readers surpass the author, just as the readers of the "Mathematical Principles of Natural Philosophy" surpassed the great Newton! Legendre was not like Newton or Euler, but his readers included two geniuses: Karl Gauss and Evariste Galois.
Such a high accuracy of geniuses was facilitated by the French Revolution, which proclaimed the state cult of Reason. After that, every talented scientist felt like Columbus or Alexander the Great, capable of discovering or conquering a new world. Many succeeded, because in the 19th century, scientific and technological progress became the main driver of the evolution of mankind, and all reasonable rulers (starting with Napoleon) were aware of this.
Gauss was close to Columbus in character. But he (like Newton) did not know how to captivate the imagination of rulers or students with beautiful speeches, and therefore limited his ambitions to the sphere of scientific concepts. Here he could do everything he wanted. For example, the ancient problem of trisection of an angle for some reason cannot be solved using a compass and a ruler. With the help of complex numbers representing points of the plane, Gauss translates this problem into the language of algebra - and obtains a general theory of the feasibility of certain geometric constructions. Thus, at the same time, a rigorous proof of the impossibility of constructing a regular 7- or 9-gon with a compass and a ruler appeared and a method of constructing a regular 17-gon that the wisest geometers of Hellas did not dream of.
Of course, such success is not for nothing: you have to invent new concepts that reflect the essence of the matter. Newton introduced three such concepts: fluxia (derivative), fluent (integral) and power series. They were enough to create mathematical analysis and the first scientific model of the physical world, including mechanics and astronomy. Gauss also introduced three new concepts: vector space, field and ring. A new algebra grew out of them, subjugating Greek arithmetic and the theory of number functions created by Newton. It still remained to subordinate algebra to the logic created by Aristotle: then it will be possible, by means of calculations, to prove the derivability or non-derivability of any scientific statements from a given set of axioms! For example, is Fermat's theorem deduced from the axioms of arithmetic, or Euclid's postulate of parallel lines - from other axioms of planimetry?
This audacious dream Gauss did not manage to realize - although he made great progress and guessed the possibility of the existence of exotic (non-commutative) algebras. Only the impudent Russian Nikolai Lobachevsky was able to construct the first non-Euclidean geometry, and the first non-commutative algebra (Group Theory) was managed by the Frenchman Evariste Galois. And only much later than Gauss's death - in 1872 - the young German Felix Klein realized that the variety of possible geometries could be brought into one-to-one correspondence with the variety of possible algebras. Simply put, every geometry is defined by its symmetry group - while general algebra studies all possible groups and their properties.
But such an understanding of geometry and algebra came much later, and the storming of Fermat's theorem was renewed even during the life of Gauss. He himself neglected Fermat's theorem out of the principle: it is not tsar's business to solve individual problems that do not fit into a vivid scientific theory! But Gauss's students, armed with his new algebra and the classical analysis of Newton and Euler, reasoned differently. First, Peter Dirichlet proved Fermat's theorem for degree 7 using the ring of complex integers generated by roots of this degree from unity. Then Ernst Kummer extended the Dirichlet method to ALL simple degrees (!) - so it seemed to him in a rage, and he triumphed. But soon a sobering up came: the proof is flawless only if every element of the ring can be uniquely decomposed into prime factors! For ordinary integers, this fact was known to Euclid, but only Gauss gave a rigorous proof. What about complex integers?
According to the "principle of the greatest mischief", there can and MUST be an ambiguous factorization! As soon as Kummer learned how to calculate the degree of ambiguity by methods of mathematical analysis, he discovered this dirty trick in the ring for the degree 23. Gauss did not have time to learn about this variant of exotic commutative algebra, but Gauss's students grew up in the place of another dirty trick a new beautiful Theory of Ideals. True, this did not particularly help the solution of Fermat's problem: its natural complexity only became clearer.
Throughout the 19th century, this ancient idol demanded more and more sacrifices from its admirers in the form of new complex theories. It is not surprising that by the beginning of the twentieth century, believers became discouraged and rebelled, rejecting their former idol. The word "fermatist" has become an abusive nickname among professional mathematicians. And although a considerable prize was awarded for the complete proof of Fermat's theorem, its applicants were mostly self-confident ignoramuses. The strongest mathematicians of that time - Poincaré and Hilbert - defiantly avoided this topic.
In 1900, Hilbert did not include Fermat's theorem in the list of twenty-three major problems facing 20th century mathematics. True, he included in their series the general problem of the solvability of Diophantine equations. The hint was clear: follow the example of Gauss and Galois, create general theories of new mathematical objects! Then one fine (but not predictable in advance) day, the old thorn will fall out by itself.
This is how the great romantic Henri Poincaré acted. Neglecting many "eternal" problems, all his life he studied the SYMMETRY of certain objects of mathematics or physics: either functions of a complex variable, or the trajectories of celestial bodies, or algebraic curves or smooth manifolds (these are multidimensional generalizations of curved lines). The motive for his actions was simple: if two different objects have similar symmetries, it means that an internal relationship is possible between them, which we are not yet able to comprehend! For example, each of the two-dimensional geometries (Euclid, Lobachevsky, or Riemann) has its own symmetry group that acts on the plane. But the points of the plane are complex numbers: in this way, the action of any geometric group is transferred to the boundless world of complex functions. It is possible and necessary to study the most symmetric of these functions: AUTOMORPHOUS (which are subject to the Euclidean group) and MODULAR (which are subject to the Lobachevsky group)!
And on the plane there are elliptical curves. They have nothing to do with the ellipse, but are given by equations of the form Y 2 \u003d AX 3 + BX 2 + CX and therefore intersect with any straight line at three points. This fact allows us to introduce multiplication among the points of an elliptic curve - to turn it into a group. The algebraic structure of this group reflects the geometric properties of the curve, maybe it is uniquely determined by its group? This question is worth studying, since for some curves the group of interest to us turns out to be modular, that is, it is related to the geometry of Lobachevsky ...
This is how Poincaré reasoned, seducing the mathematical youth of Europe, but at the beginning of the twentieth century these temptations did not lead to vivid theorems or hypotheses. It turned out differently with Hilbert's appeal: to study general solutions of Diophantine equations with integer coefficients! In 1922, a young American Lewis Mordell connected the set of solutions of such an equation (this is a vector space of a certain dimension) with the geometric genus of the complex curve that is given by this equation. Mordell came to the conclusion that if the degree of the equation is large enough (more than two), then the dimension of the solution space is expressed through the genus of the curve, and therefore this dimension is FINITE. On the contrary - to the power of 2, the Pythagorean equation has an INFINITE family of solutions!
Of course, Mordell saw the connection between his hypothesis and Fermat's theorem. If it becomes known that for each degree n\u003e 2 the space of entire solutions of Fermat's equation is finite-dimensional, this will help to prove that there are no such solutions at all! But Mordell did not see any ways to prove his hypothesis - and although he lived long life, but did not wait for the transformation of this hypothesis into the Faltings theorem. This happened in 1983 - in a completely different era, after the great successes of the algebraic topology of varieties.
Poincaré created this science as if by accident: he wanted to know what three-dimensional varieties are. After all, Riemann figured out the structure of all closed surfaces and received a very simple answer! If there is no such answer in the three-dimensional or multidimensional case, you need to come up with a system of algebraic invariants of the manifold that determines its geometric structure. It is best if such invariants are elements of some groups - commutative or non-commutative.
Oddly enough, this daring Poincaré plan succeeded: it was carried out from 1950 to 1970 thanks to the efforts of so many geometers and algebraists. Until 1950, there was a quiet accumulation of various methods of classifying varieties, and after that date a critical mass of people and ideas seemed to accumulate and an explosion burst out, comparable to the invention of mathematical analysis in the 17th century. But the analytical revolution stretched out for a century and a half, encompassing the creative biographies of four generations of mathematicians - from Newton and Leibniz to Fourier and Cauchy. On the contrary, the topological revolution of the twentieth century was completed in twenty years - thanks to the large number of its participants. At the same time, a large generation of self-confident young mathematicians has developed, who suddenly found themselves without work in their historical homeland.
In the seventies, they rushed to the adjacent areas of mathematics and theoretical physics. Many have established their scientific schools in dozens of universities in Europe and America. Many students of different ages and nationalities, with different abilities and inclinations, still circulate between these centers, and everyone wants to be famous for some discovery. It was in this confusion that Mordell's conjecture and Fermat's theorem were finally proved.
However, the first swallow, unaware of its fate, grew up in Japan in the hungry and unemployed post-war years. The name of the swallow was Yutaka Taniyama. In 1955, this hero turned 28 years old, and he decided (together with friends Goro Shimura and Takauji Tamagawa) to revive mathematical research in Japan. Where to begin? Of course, with overcoming isolation from foreign colleagues! So in 1955, three young Japanese people organized the first international conference on algebra and number theory in Tokyo. It was apparently easier to do this in Japan, re-educated by the Americans, than in Russia frozen by Stalin ...
Among the guests of honor were two heroes from France: André Weil and Jean-Pierre Serre. Here the Japanese were very lucky: Weil was the recognized head of the French algebraists and a member of the Bourbaki group, and the young Serre played a similar role among topologists. In heated discussions with them, the heads of Japanese youth cracked, their brains melted, but as a result, such ideas and plans crystallized that could hardly have been born in another environment.
One day Taniyama came to Weyl with a question about elliptic curves and modular functions. At first, the Frenchman did not understand anything: Taniyama was not a master of speaking in English. Then the essence of the matter became clear, but Taniyama did not manage to give his hopes an exact formulation. All Weil could answer to the young Japanese was that if he was very lucky in terms of inspiration, then something useful would grow out of his vague hypotheses. But so far there is little hope for that!
Obviously, Weil did not notice the heavenly fire in Taniyama's gaze. And there was a fire: it seems that for a moment the indomitable thought of the late Poincaré entered the Japanese man! Taniyama came to the conviction that every elliptic curve is generated by modular functions - more precisely, it is "uniformized by a modular form." Alas, this exact formulation was born much later - in conversations between Taniyama and his friend Shimura. And then Taniyama committed suicide in a fit of depression ... His hypothesis was left without a master: it was not clear how to prove it or where to test it, and therefore no one took it seriously for a long time. The first response came only thirty years later - almost like in the era of Fermat!
The ice broke in 1983 when the twenty-seven-year-old German Gerd Faltings announced to the whole world: Mordell's hypothesis was proved! Mathematicians were wary, but Faltings was a true German: there were no gaps in his long and complex proof. It's just that the time has come, facts and concepts have accumulated - and now one talented algebraist, relying on the results of ten other algebraists, managed to solve a problem that had been waiting for the owner for sixty years. This is not uncommon in 20th century mathematics. It is worth recalling the secular continuum problem in set theory, the two Burnside conjectures in group theory, or the Poincaré conjecture in topology. Finally, in the theory of numbers, the time has come to reap the harvest of old crops ... What peak will be the next in the line of conquered by mathematicians? Will Euler's problem, Riemann's hypothesis, or Fermat's theorem collapse? It would be good!
And now, two years after Faltings' revelation, another inspired mathematician appeared in Germany. His name was Gerhard Frey, and he stated something strange: as if Fermat's theorem was deduced from Taniyama's hypothesis! Unfortunately, Frey's style of presenting his thoughts was more reminiscent of the unlucky Taniyama than of his articulate compatriot Faltings. In Germany, no one understood Frey, and he went overseas - to the glorious town of Princeton, where after Einstein they got used to not such visitors. No wonder Barry Mazur made his nest there - a versatile topologist, one of the heroes of the recent assault on smooth manifolds. And a pupil, Ken Ribet, who was equally experienced in the intricacies of topology and algebra, but who had not yet glorified himself, grew up next to Mazur.
Having heard Frey's speech for the first time, Ribet decided that it was nonsense and pseudo-scientific fiction (probably, Weil reacted in the same way to Taniyama's revelations). But Ribet could not forget this “fantasy” and at times returned to it mentally. Six months later, Ribet believed that there was something useful in Frey's fantasies, and a year later he decided that he himself could almost prove Frey's strange hypothesis. But some "holes" remained, and Ribet decided to confess to his boss Mazur. The latter listened attentively to the student and calmly replied: “You have done everything! Here you need to apply the transformation Ф, here - use Lemmas B and K, and everything will take a flawless form! " So Ribet made the leap from obscurity to immortality, using a catapult in the person of Frey and Mazur. In all fairness, all of them - along with the late Taniyama - should be considered as the proofs of Fermat's Great Theorem.
But the trouble is: they deduced their statement from Taniyama's hypothesis, which itself has not been proven! What if she's wrong? Mathematicians have long known that "anything follows from a lie," if Taniyama's guess is wrong, then Ribet's impeccable reasoning is worthless! There is an urgent need to prove (or disprove) Taniyama's conjecture - otherwise someone like Faltings will prove Fermat's theorem in a different way. He will be a hero!
It is unlikely that we will ever know how many young or seasoned algebraists pounced on Fermat's theorem after the success of Faltings or after the victory of Ribet in 1986. All of them tried to work in secret, so that in case of failure they would not be reckoned among the community of "dummies" - fermatists. It is known that the luckiest of all - Andrew Wiles from Cambridge - only got the taste of victory at the beginning of 1993. This not so much rejoiced as it scared Wiles: what if there was a mistake or a gap in his proof of Taniyama's hypothesis? Then his scientific reputation perished! You have to carefully write down the proof (but it will be many tens of pages!) And postpone it for six months or a year, so that you can then reread it coolly and meticulously ... But if during this time someone publishes their proof? Oh, trouble ...
Yet Wiles came up with a double way to quickly test his proof. First, you need to trust one of your trusted friends and colleagues and tell him the whole line of reasoning. From the outside all the mistakes are better known! Secondly, it is necessary to read a special course on this topic to smart students and graduate students: these smart guys will not miss a single mistake of the lecturer! Just do not tell them the ultimate goal of the course until the last moment - otherwise the whole world will know about it! And of course, you need to look for such an audience farther from Cambridge - better not even in England, but in America ... What could be better than distant Princeton?
This is where Wiles went in the spring of 1993. His patient friend Niklas Katz, after listening to Wiles's long report, found a number of gaps in it, but they all turned out to be easily fixable. But the Princeton graduate students soon fled from Wiles' special course, not wanting to follow the whimsical thought of the lecturer, who leads them to no one knows where. After this (not particularly deep) examination of his work, Wiles decided it was time to bring a great miracle to the world.
In June 1993, a regular conference was held in Cambridge on "Iwasawa theory" - a popular branch of number theory. Wiles decided to share his proof of Taniyama's hypothesis on it without announcing main result until the very end. The report went on for a long time, but it was successful, gradually journalists began to flock, who sensed something. Finally, thunder struck: Fermat's theorem is proved! The general jubilation was not overshadowed by any doubts: it seems that everything is clear ... But after two months Katz, after reading Wiles's final text, noticed another gap in it. Some transition in reasoning was based on the "Euler system" - but what Wiles built was not such a system!
Wiles checked the bottleneck and realized he was wrong. Even worse: it is not clear how to replace erroneous reasoning! After that, Wiles's life fell into the darkest months. Previously, he freely synthesized an unprecedented proof from the material at hand. Now he is tied to a narrow and precise task - without the confidence that it has a solution and that he will be able to find it in the foreseeable future. Recently Frey could not resist the same struggle - and now his name was obscured by the name of the successful Ribet, although Frey's guess turned out to be correct. And what will happen to MY guess and to MY name?
This hard labor dragged on for exactly a year. In September 1994, Wiles was ready to admit defeat and leave Taniyama's hypothesis to more fortunate successors. Having made this decision, he slowly began to reread his proof - from beginning to end, listening to the rhythm of reasoning, reliving the pleasure of successful finds. When he reached the "damned" place, Wiles, however, did not hear the false note in his mind. Was the course of his reasoning flawless after all, and the mistake arose only in the WORDING description of the mental image? If there is no "Euler system" here, then what is hidden here?
Suddenly, a simple thought came up: the "Euler system" does not work where the Iwasawa theory is applicable. Why not apply this theory directly - fortunately, Wiles himself is familiar and familiar with it? And why didn't he try this approach from the very beginning, but got carried away by someone else's vision of the problem? Wiles couldn’t recall these details, and it was useless. He did the necessary reasoning within the framework of Iwasawa's theory, and everything worked out in half an hour! So - with a delay of one year - the last gap in the proof of Taniyama's hypothesis was closed. The final text was given to be torn apart by a group of reviewers of the famous mathematical journal, a year later they announced that now there are no mistakes. Thus, in 1995, Fermat's last hypothesis passed away at the three hundred and sixtieth year of her life, becoming a proven theorem, which will inevitably enter the textbooks of number theory.
Summing up the three-century fuss over Fermat's theorem, we have to draw a strange conclusion: this heroic epic might not have happened! Indeed, the Pythagorean theorem expresses a simple and important connection between visual natural objects - the lengths of the segments. But the same cannot be said about Fermat's theorem. It looks more like a cultural superstructure on a scientific substrate - like reaching the Earth's North Pole or flying to the moon. Let us recall that both of these feats were sung by writers long before their accomplishment - back in ancient times, after the appearance of Euclid's "Principles", but before the appearance of Diophantus's "Arithmetic". This means that then a social need arose for intellectual feats of this sort - at least imaginary! Before the Hellenes had enough of Homer's poems, just as a hundred years before Fermat, the French had enough religious hobbies. But then religious passions subsided - and science stood next to them.
In Russia, such processes began a hundred and fifty years ago, when Turgenev put Yevgeny Bazarov on a par with Yevgeny Onegin. True, the writer Turgenev did not understand well the motives of the actions of the scientist Bazarov and did not dare to sing them, but this was soon done by the scientist Ivan Sechenov and the enlightened journalist Jules Verne. The spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and then science fiction appears first, and then popular science literature (including the magazine "Knowledge is Power").
At the same time, a specific scientific topic is not at all important for the general public and is not very important even for the performer heroes. So, hearing about the reaching of the North Pole by Piri and Cook, Amundsen instantly changed the goal of his already prepared expedition - and soon reached the South Pole, ahead of Scott by one month. Later, the successful flight of Yuri Gagarin around the Earth forced President Kennedy to change the previous goal of the American space program for a more expensive, but much more impressive: landing of people on the moon.
Even earlier, the shrewd Hilbert answered the naive question of students: "The solution of what scientific problem would be most useful now?" - answered jokingly: "Catch a fly on the far side of the moon!" To the bewildered question: "Why is this necessary?" - followed by a clear answer: “THIS is not needed by anyone! But think about the scientific methods and technical means that we will have to develop to solve such a problem - and how many other beautiful problems we will solve along the way! "
This is exactly what happened with Fermat's theorem. Euler may well have missed her.
In this case, some other problem would become the idol of mathematicians - perhaps also from number theory. For example, the problem of Eratosthenes: finite or infinite prime numbers- twins (such as 11 and 13, 17 and 19, and so on)? Or Euler's problem: is every even number the sum of two primes? Or: is there an algebraic relationship between the numbers π and e? These three problems have not yet been resolved, although in the twentieth century mathematicians have noticeably come closer to understanding their essence. But this century also gave rise to many new, no less interesting problems, especially at the junctions of mathematics with physics and other branches of natural science.
Back in 1900, Hilbert singled out one of them: to create a complete system of axioms of mathematical physics! A hundred years later, this problem is far from being solved - if only because the arsenal of mathematical tools in physics is steadily growing, and not all of them have a rigorous justification. But after 1970, theoretical physics split into two branches. One (classical) since the time of Newton has been engaged in modeling and forecasting SUSTAINABLE processes, the other (newborn) is trying to formalize the interaction of UNSTABLE processes and ways to control them. It is clear that these two branches of physics must be axiomatized separately.
The first of them will probably be able to cope with in twenty or fifty years ...
And what is lacking in the second branch of physics - the one that deals with all kinds of evolution (including outlandish fractals and strange attractors, the ecology of biocenoses and Gumilev's theory of passionarity)? We will hardly understand this soon. But the worship of scientists to a new idol has already become a mass phenomenon. Probably, an epic will unfold here, comparable to the three-century biography of Fermat's theorem. So, at the junction of different sciences, more and more new idols are born - similar to religious ones, but more complex and dynamic ...
Apparently, a person cannot remain a person without overthrowing old idols from time to time and not creating new ones - in torment and with joy! Pierre Fermat was lucky to be in a fateful moment near the hot spot of the birth of a new idol - and he managed to leave an imprint of his personality on the newborn. One can envy such a fate, and it is not a sin to imitate it.
Sergey Smirnov
"Knowledge is power"
Ivliev 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, but 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, we will 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 the Euclidean space, instead of (1) and (3) we obtain formulas that are very 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, nevertheless the most important criterion is to be an equation of the third degree in linear system it does not satisfy coordinates.
b) Error classification
So, once again, let's return to the beginning of the consideration and trace how it is concluded that the WTF is true. First, it is assumed that there is a solution to Fermat's equation in positive integers. Second, 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 to Fermat's equation is simultaneously a solution algebraic equation 3rd degree, 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 Fermat's equation is compared (apparently, presumably unambiguously) with a fictitious, nonexistent elliptic curve, and then all the pathos of further reasoning is spent on proving that a specific elliptic curve of this type, obtained from hypothetical solutions of Fermat's 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 specified type. 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, her equation C 2 \u003d A 2 + B 2 has no integer solutions for integers x, y, z and n ≥ 3. In nonlinear coordinate axes X and Y, such a circle would be described by the equation appearance very similar to the standard form:
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 is immediately striking. 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 to 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 the form (5) can be represented as a decomposition into prime factors of complex numbers similar to the algebraic identity (6), but 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 Frey voiced his witty idea. The consequences of such an imprudent statement brought mathematics as a whole to the brink of losing public trust, which is described in detail in and which necessarily raises the question of responsibility for science. scientific institutions in front of society. Comparison of Fermat's equation with Frey's curve (1) is the "lock" of the whole proof of Wiles regarding Fermat's theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof either.
IN recent times there are various Internet reports 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 - United science Magazine (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 # 3 p.443-551.
Bibliographic reference
Ivliev Yu.A. WYLES 'MISTAKE 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: 03.03.2020). We bring to your attention the journals published by the "Academy of Natural Sciences"