The History of Farm's Great Theorem. Felix Kirsanov
The Grand Theorem Farm Singh Simon
"Has Fermat's Last Theorem been proven?"
It was only the first step towards proving the Taniyama-Shimura conjecture, but the strategy chosen by Wiles was a brilliant mathematical breakthrough, a result that deserved to be published. But due to the vow of silence imposed by Wiles on himself, he could not tell the rest of the world about the result and had no idea who else could make such a significant breakthrough.
Wiles recalls his philosophical attitude towards any potential challenger: “No one wants to spend years proving something and find that someone else managed to find the proof a few weeks earlier. But, oddly enough, since I was trying to solve a problem that was essentially considered insoluble, I was not very afraid of my opponents. I just didn't expect myself or anyone else to come up with an idea that would lead to a proof."
On March 8, 1988, Wiles was shocked to see front-page headlines in large print that read: "Fermat's Last Theorem Proven." The Washington Post and the New York Times reported that 38-year-old Yoichi Miyaoka of Tokyo Metropolitan University had solved the world's most difficult mathematical problem. So far, Miyaoka has not yet published his proof, but he outlined its course at a seminar at the Max Planck Institute for Mathematics in Bonn. Don Zagier, who attended Miyaoka's report, expressed the optimism of the mathematical community in the following words: “The proof presented by Miyaoka is extremely interesting, and some mathematicians believe that it will turn out to be correct with a high probability. There is no certainty yet, but so far the evidence looks very encouraging.”
Speaking at a seminar in Bonn, Miyaoka spoke about his approach to solving the problem, which he considered from a completely different, algebro-geometric, point of view. Over the past decades, geometers have achieved a deep and subtle understanding of mathematical objects, in particular, the properties of surfaces. In the 1970s, the Russian mathematician S. Arakelov tried to establish parallels between problems in algebraic geometry and problems in number theory. This was one of the lines of Langlands' program, and mathematicians hoped that unsolved problems in number theory could be solved by studying the corresponding problems in geometry, which also remained unsolved. Such a program was known as the philosophy of concurrency. Those algebraic geometers who tried to solve problems in number theory were called "arithmetic algebraic geometers". In 1983, they heralded their first significant victory when Gerd Faltings of the Princeton Institute for Advanced Study made significant contributions to the understanding of Fermat's Theorem. Recall that, according to Fermat, the equation
at n greater than 2 has no solutions in integers. Faltings thought he had made progress in proving Fermat's Last Theorem by studying the geometric surfaces associated with different values n. Surfaces associated with Fermat's equations for various values n, differ from each other, but have one common property - they all have through holes, or, simply speaking, holes. These surfaces are four-dimensional, as are the graphs of modular forms. Two-dimensional sections of two surfaces are shown in fig. 23. The surfaces associated with Fermat's equation look similar. The greater the value n in the equation, the more holes in the corresponding surface.
Rice. 23. These two surfaces were obtained using the computer program Mathematica. Each of them represents the locus of points satisfying the equation x n + y n = z n(for the surface on the left n=3, for the surface on the right n=5). Variables x And y are considered to be complex.
Faltings was able to prove that, since such surfaces always have several holes, the associated Fermat equation could only have a finite set of solutions in integers. The number of solutions could be anything from zero, as Fermat suggested, to a million or a billion. Thus, Faltings did not prove Fermat's Last Theorem, but at least managed to reject the possibility that Fermat's equation could have infinitely many solutions.
Five years later, Miyaoka reported that he had gone one step further. He was then in his early twenties. Miyaoka formulated a conjecture about some inequality. It became clear that proving his geometric conjecture would mean proving that the number of solutions to Fermat's equation is not only finite, but zero. Miyaoka's approach was similar to Wiles' in that they both tried to prove Fermat's Last Theorem by relating it to a fundamental conjecture in another area of mathematics. For Miyaoka it was algebraic geometry, for Wiles the path to proof lay through elliptic curves and modular forms. Much to Wiles' dismay, he was still struggling with the proof of the Taniyama-Shimura conjecture when Miyaoka claimed to have a complete proof of his own conjecture, and therefore of Fermat's Last Theorem.
Two weeks after his speech in Bonn, Miyaoka published the five pages of calculations that formed the essence of his proof, and a thorough check began. Number theorists and algebraic geometries all over the world studied, line by line, published calculations. A few days later, mathematicians discovered one contradiction in the proof, which could not but cause concern. One part of Miyaoka's work led to a statement from number theory, from which, when translated into the language of algebraic geometry, a statement was obtained that contradicted the result obtained several years earlier. While this did not necessarily invalidate Miyaoka's entire proof, the discrepancy that was discovered did not fit into the philosophy of parallelism between number theory and geometry.
Two weeks later, Gerd Faltings, who paved the way for Miyaoke, announced that he had discovered the exact cause of the apparent violation of concurrency - a gap in reasoning. The Japanese mathematician was a geometer and was not absolutely strict in translating his ideas into the less familiar territory of number theory. An army of number theorists made desperate efforts to patch up the hole in Miyaoki's proof, but in vain. Two months after Miyaoka announced that he had a complete proof of Fermat's Last Theorem, the mathematical community came to the unanimous conclusion that Miyaoka's proof was doomed to failure.
As in the case of previous failed proofs, Miyaoka managed to obtain many interesting results. Parts of his proof deserve attention as very ingenious applications of geometry to number theory, and in later years other mathematicians used them to prove certain theorems, but no one succeeded in proving Fermat's Last Theorem in this way.
The hype about Fermat's Last Theorem soon died down, and the newspapers carried brief notes saying that the three-hundred-year-old puzzle still remained unsolved. On the wall of the New York subway station on Eighth Street appeared the following inscription, no doubt inspired by press publications about Fermat's Last Theorem: "The equation xn + yn = zn has no solutions. I have found a truly amazing proof of this fact, but I cannot write it down here because my train has come.
CHAPTER 10 CROCODILE FARM They drove along the scenic road in old John's car, sitting in the back seats. Behind the wheel was a black driver in a brightly colored shirt with an oddly cropped head. Bushes of black hair, hard as wire, rose on a shaved skull, logic
Race preparation. Alaska, Linda Pletner's Iditarod Farm is an annual dog sled race in Alaska. The length of the route is 1150 miles (1800 km). This is the longest dog sled race in the world. Start (ceremonial) - March 4, 2000 from Anchorage. Start
Goat Farm There is a lot of work in the village during the summer. When we visited the village of Khomutets, hay was being harvested and fragrant waves from freshly cut grass seemed to soak everything around. Grasses must be mowed in time so that they do not overripe, then everything valuable and nutritious will be preserved in them. This
Summer farm Straw, like lightning hand, into glass grass Another, having signed on the fence, lit the fire of the green glass of Water in the horse's trough. Into the blue dusk Wander, swaying, nine ducks along the rut of the spirit of parallel lines. Here is a chicken staring at nothing alone
Ruined farm The calm sun, like a flower of dark red, Went down to the earth, growing into the sunset, But the curtain of the night in idle power Twitched the world, which disturbed the look. Silence reigned on a farm without a roof, As if someone had torn off her hair, They fought over a cactus
Farm or backyard? On February 13, 1958, all the central Moscow and then regional newspapers published the decision of the Central Committee of the Communist Party of Ukraine "On an error in the purchase of cows from collective farmers in the Zaporozhye region." It was not even about the entire region, but about two of its districts: Primorsky
Fermat's problem In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. “At school, I loved solving problems, I took them home and came up with new ones from each problem. But the best problem I've ever come across, I found in a local
From the Pythagorean Theorem to Fermat's Last Theorem The Pythagorean theorem and the infinite number of Pythagorean triples were discussed in the book by E.T. Bell's "The Great Problem" - the same library book that caught the attention of Andrew Wiles. And although the Pythagoreans reached almost complete
Mathematics after the proof of Fermat's Last Theorem Oddly enough, Wiles himself had mixed feelings about his report: “The occasion for the speech was very well chosen, but the lecture itself aroused mixed feelings in me. Work on the proof
CHAPTER 63 Old McLennon's Farm About a month and a half after returning to New York on one of the "November evenings" the phone rang at the Lennons' apartment. Yoko picked up the phone. A Puerto Rican male voice asked Yoko Ono.
Pontryagin's theorem Simultaneously with the Conservatory, dad studied at Moscow State University, at the Mechanics and Mathematics. He successfully graduated from it and even hesitated for some time in choosing a profession. Musicology won, as a result of which he benefited from his mathematical mindset. One of my father's fellow students
Theorem The theorem on the right of a religious association to choose a priest needs to be proved. It reads like this: "An Orthodox community is being created... under the spiritual guidance of a priest chosen by the community and having received the blessing of the diocesan bishop."
I. Farm (“Here, from chicken manure…”) Here, from chicken manure One salvation is a broom. Love - which counts? - They took me to the chicken coop. Pecking at the grain, the hens cackle, the roosters march importantly. And without size and censorship Poems are composed in the mind. About a Provençal afternoon
Pierre de Fermat, reading the "Arithmetic" of Diophantus of Alexandria and reflecting on its problems, had the habit of writing down the results of his reflections in the form of brief remarks in the margins of the book. Against the eighth problem of Diophantus in the margins of the book, Fermat wrote: " On the contrary, it is impossible to decompose neither a cube into two cubes, nor a bi-square into two bi-squares, and, in general, no degree greater than a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, but these margins are too narrow for it.» / E.T.Bell "Creators of Mathematics". M., 1979, p.69/. I bring to your attention an elementary proof of the farm theorem, which can be understood by any high school student who is fond of mathematics.
Let us compare Fermat's commentary on the Diophantine problem with the modern formulation of Fermat's great theorem, which has the form of an equation.
« The equation
x n + y n = z n(where n is an integer greater than two)
has no solutions in positive integers»
The comment is in a logical connection with the task, similar to the logical connection of the predicate with the subject. What is affirmed by the problem of Diophantus, on the contrary, is affirmed by Fermat's commentary.
Fermat's comment can be interpreted as follows: if a quadratic equation with three unknowns has an infinite number of solutions on the set of all triples of Pythagorean numbers, then, on the contrary, an equation with three unknowns in a degree greater than the square
There is not even a hint of its connection with the Diophantine problem in the equation. His assertion requires proof, but it does not have a condition from which it follows that it has no solutions in positive integers.
The variants of the proof of the equation known to me are reduced to the following algorithm.
- The equation of Fermat's theorem is taken as its conclusion, the validity of which is verified with the help of proof.
- The same equation is called initial the equation from which its proof must proceed.
The result is a tautology: If an equation has no solutions in positive integers, then it has no solutions in positive integers.". The proof of the tautology is obviously wrong and devoid of any meaning. But it is proved by contradiction.
- An assumption is made that is the opposite of that stated by the equation to be proved. It should not contradict the original equation, but it does. To prove what is accepted without proof, and to accept without proof what is required to be proved, does not make sense.
- Based on the accepted assumption, absolutely correct mathematical operations and actions are performed to prove that it contradicts the original equation and is false.
Therefore, for 370 years now, the proof of the equation of Fermat's Last Theorem has remained an impossible dream of specialists and lovers of mathematics.
I took the equation as the conclusion of the theorem, and the eighth problem of Diophantus and its equation as the condition of the theorem.
"If the equation x2 + y2 = z2
(1) has an infinite set of solutions on the set of all triples of Pythagorean numbers, then, conversely, the equation x n + y n = z n
, Where n > 2
(2) has no solutions on the set of positive integers."
Proof.
A) Everyone knows that equation (1) has an infinite number of solutions on the set of all triples of Pythagorean numbers. Let us prove that no triple of Pythagorean numbers, which is a solution to equation (1), is a solution to equation (2).
Based on the law of reversibility of equality, the sides of equation (1) are interchanged. Pythagorean numbers (z, x, y) can be interpreted as the lengths of the sides of a right triangle, and the squares (x2, y2, z2) can be interpreted as the areas of squares built on its hypotenuse and legs.
We multiply the squares of equation (1) by an arbitrary height h :
z 2 h = x 2 h + y 2 h (3)
Equation (3) can be interpreted as the equality of the volume of a parallelepiped to the sum of the volumes of two parallelepipeds.
Let the height of three parallelepipeds h = z :
z 3 = x 2 z + y 2 z (4)
The volume of the cube is decomposed into two volumes of two parallelepipeds. We leave the volume of the cube unchanged, and reduce the height of the first parallelepiped to x and the height of the second parallelepiped will be reduced to y . The volume of a cube is greater than the sum of the volumes of two cubes:
z 3 > x 3 + y 3 (5)
On the set of triples of Pythagorean numbers ( x, y, z ) at n=3 there can be no solution to equation (2). Consequently, on the set of all triples of Pythagorean numbers, it is impossible to decompose a cube into two cubes.
Let in equation (3) the height of three parallelepipeds h = z2 :
z 2 z 2 = x 2 z 2 + y 2 z 2 (6)
The volume of a parallelepiped is decomposed into the sum of the volumes of two parallelepipeds.
We leave the left side of equation (6) unchanged. On its right side the height z2
reduce to X
in the first term and up to at 2
in the second term.
Equation (6) turned into the inequality:
The volume of a parallelepiped is decomposed into two volumes of two parallelepipeds.
We leave the left side of equation (8) unchanged.
On the right side of the height zn-2
reduce to xn-2
in the first term and reduce to y n-2
in the second term. Equation (8) turns into the inequality:
| z n > x n + y n | (9) |
On the set of triples of Pythagorean numbers, there cannot be a single solution of equation (2).
Consequently, on the set of all triples of Pythagorean numbers for all n > 2 equation (2) has no solutions.
Obtained "post miraculous proof", but only for triplets Pythagorean numbers. This is lack of evidence and the reason for the refusal of P. Fermat from him.
b) Let us prove that equation (2) has no solutions on the set of triples of non-Pythagorean numbers, which is the family of an arbitrarily taken triple of Pythagorean numbers z=13, x=12, y=5 and the family of an arbitrary triple of positive integers z=21, x=19, y=16
Both triplets of numbers are members of their families:
| (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 the product of 13 by 12 and 21 by 20, i.e. 78 and 210.
Each member of the family (10) contains z = 13 and variables X And at 13 > x > 0 , 13 > y > 0 1
Each member of the family (11) contains z = 21 and variables X And at , which take integer values 21 > x >0 , 21 > y > 0 . The variables decrease sequentially by 1 .
The triples of numbers of the sequence (10) and (11) can be represented as a sequence of inequalities of the third degree:
| 13 3 < 12 3 + 12 3 ;13 3 < 12 3 + 11 3 ;…; 13 3 < 12 3 + 8 3 ; 13 3 > 12 3 + 7 3 ;…; 13 3 > 1 3 + 1 3 | ||
| 21 3 < 20 3 + 20 3 ; 21 3 < 20 3 + 19 3 ; …; 21 3 < 19 3 + 14 3 ; 21 3 > 19 3 + 13 3 ;…; 21 3 > 1 3 + 1 3 |
and in the form of inequalities of the fourth degree:
| 13 4 < 12 4 + 12 4 ;…; 13 4 < 12 4 + 10 4 ; 13 4 > 12 4 + 9 4 ;…; 13 4 > 1 4 + 1 4 | ||
| 21 4 < 20 4 + 20 4 ; 21 4 < 20 4 + 19 4 ; …; 21 4 < 19 4 + 16 4 ;…; 21 4 > 1 4 + 1 4 |
The correctness of each inequality is verified by raising the numbers to the third and fourth powers.
The cube of a larger number cannot be decomposed into two cubes of smaller numbers. It is either less than or greater than the sum of the cubes of the two smaller numbers.
The bi-square of a larger number cannot be decomposed into two bi-squares of smaller numbers. It is either less than or greater than the sum of the bi-squares of smaller numbers.
As the exponent increases, all inequalities, except for the leftmost inequality, have the same meaning:
Inequalities, they all have the same meaning: the degree of the larger number is greater than the sum of the degrees of the smaller two numbers with the same exponent:
| 13n > 12n + 12n ; 13n > 12n + 11n ;…; 13n > 7n + 4n ;…; 13n > 1n + 1n | (12) | |
| 21n > 20n + 20n ; 21n > 20n + 19n ;…; ;…; 21n > 1n + 1n | (13) |
The leftmost term of sequences (12) (13) is the weakest inequality. Its correctness determines the correctness of all subsequent inequalities of the sequence (12) for n > 8 and sequence (13) for n > 14 .
There can be no equality among them. An arbitrary triple of positive integers (21,19,16) is not a solution to equation (2) of Fermat's Last Theorem. If an arbitrary triple of positive integers is not a solution to the equation, then the equation has no solutions on the set of positive integers, which was to be proved.
WITH) Fermat's commentary on the Diophantus problem states that it is impossible to decompose " in general, no power greater than the square, two powers with the same exponent».
Kiss a power greater than a square cannot really be decomposed into two powers with the same exponent. I don't kiss a power greater than the square can be decomposed into two powers with the same exponent.
Any randomly chosen triple of positive integers (z, x, y) may belong to a family, each member of which consists of a constant number z and two numbers less than z . Each member of the family can be represented in the form of an inequality, and all the resulting inequalities can be represented as a sequence of inequalities:
| z n< (z — 1) n + (z — 1) n ; z n < (z — 1) n + (z — 2) n ; …; z n >1n + 1n | (14) |
The sequence of inequalities (14) begins with inequalities whose left side is less than the right side and ends with inequalities whose right side is less than the left side. With increasing exponent n > 2 the number of inequalities on the right side of sequence (14) increases. With an exponent n=k all the inequalities on the left side of the sequence change their meaning and take on the meaning of the inequalities on the right side of the inequalities of the sequence (14). As a result of the increase in the exponent of all inequalities, the left side is greater than the right side:
| z k > (z-1) k + (z-1) k ; z k > (z-1) k + (z-2) k ;…; zk > 2k + 1k ; zk > 1k + 1k | (15) |
With a further increase in the exponent n>k none of the inequalities changes its meaning and does not turn into equality. On this basis, it can be argued that any arbitrarily taken triple of positive integers (z, x, y) at n > 2 , z > x , z > y
In an arbitrary triple of positive integers z can be an arbitrarily large natural number. For all natural numbers not greater than z , Fermat's Last Theorem is proved.
D) No matter how big the number z , in the natural series of numbers before it there is a large but finite set of integers, and after it there is an infinite set of integers.
Let us prove that the entire infinite set of natural numbers greater than z , form triples of numbers that are not solutions to the equation of Fermat's Last Theorem, for example, an arbitrary triple of positive integers (z+1,x,y) , wherein z + 1 > x And z + 1 > y for all values of the exponent n > 2 is not a solution to the equation of Fermat's Last Theorem.
A randomly chosen triple of positive integers (z + 1, x, y) may belong to a family of triples of numbers, each member of which consists of a constant number z + 1 and two numbers X And at , taking different values, smaller z + 1 . Family members can be represented as inequalities whose constant left side is less than, or greater than, the right side. The inequalities can be arranged in order as a sequence of inequalities:
With a further increase in the exponent n>k to infinity, none of the inequalities in the sequence (17) changes its meaning and does not become an equality. In sequence (16), the inequality formed from an arbitrarily taken triple of positive integers (z + 1, x, y) , can be in its right side in the form (z + 1) n > x n + y n or be on its left side in the form (z+1)n< x n + y n .
In any case, the triple of positive integers (z + 1, x, y) at n > 2 , z + 1 > x , z + 1 > y in sequence (16) is an inequality and cannot be an equality, i.e., it cannot be a solution to the equation of Fermat's Last Theorem.
It is easy and simple to understand the origin of the sequence of power inequalities (16), in which the last inequality of the left side and the first inequality of the right side are inequalities of the opposite sense. On the contrary, it is not easy and difficult for schoolchildren, high school students and high school students to understand how a sequence of inequalities (17) is formed from a sequence of inequalities (16), in which all inequalities have the same meaning.
In sequence (16), increasing the integer degree of inequalities by 1 turns the last inequality on the left side into the first inequality of the opposite meaning on the right side. Thus, the number of inequalities on the ninth side of the sequence decreases, while the number of inequalities on the right side increases. Between the last and first power inequalities of the opposite meaning, there is a power equality without fail. Its degree cannot be an integer, since there are only non-integer numbers between two consecutive natural numbers. The power equality of a non-integer degree, according to the condition of the theorem, cannot be considered a solution to equation (1).
If in the sequence (16) we continue to increase the degree by 1 unit, then the last inequality of its left side will turn into the first inequality of the opposite meaning of the right side. As a result, there will be no inequalities on the left side and only inequalities on the right side, which will be a sequence of increasing power inequalities (17). A further increase in their integer degree by 1 unit only strengthens its power inequalities and categorically excludes the possibility of the appearance of equality in an integer degree.
Therefore, in general, no integer power of a natural number (z+1) of the sequence of power inequalities (17) can be decomposed into two integer powers with the same exponent. Therefore, equation (1) has no solutions on an infinite set of natural numbers, which was to be proved.
Therefore, Fermat's Last Theorem is proved in all generality:
- in section A) for all triplets (z, x, y) Pythagorean numbers (Fermat's discovery is a truly miraculous proof),
- in section C) for all members of the family of any triple (z, x, y) pythagorean numbers,
- in section C) for all triplets of numbers (z, x, y) , not large numbers z
- in section D) for all triples of numbers (z, x, y) natural series of numbers.
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Changes were made on 05.09.2010 |
Which theorems can and which cannot be proven by contradiction
The Explanatory Dictionary of Mathematical Terms defines proof by contradiction of a theorem opposite to the inverse theorem.
“Proof by contradiction is a method of proving a theorem (sentence), which consists in proving not the theorem itself, but its equivalent (equivalent), opposite inverse (reverse to opposite) theorem. Proof by contradiction is used whenever the direct theorem is difficult to prove, but the opposite inverse is easier. When proving by contradiction, the conclusion of the theorem is replaced by its negation, and by reasoning one arrives at the negation of the condition, i.e. to a contradiction, to the opposite (the opposite of what is given; this reduction to absurdity proves the theorem.
Proof by contradiction is very often used in mathematics. The proof by contradiction is based on the law of the excluded middle, which consists in the fact that of the two statements (statements) A and A (negation of A), one of them is true and the other is false./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.112/.
It would not be better to openly declare that the method of proof by contradiction is not a mathematical method, although it is used in mathematics, that it is a logical method and belongs to logic. Is it valid to say that proof by contradiction is "used whenever a direct theorem is difficult to prove", when in fact it is used if, and only if, there is no substitute for it.
The characteristic of the relationship between the direct and inverse theorems also deserves special attention. “An inverse theorem for a given theorem (or to a given theorem) is a theorem in which the condition is the conclusion, and the conclusion is the condition of the given theorem. This theorem in relation to the converse theorem is called the direct theorem (initial). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and inverse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If the diagonals in a quadrilateral are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e., the converse theorem is not true./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.261 /.
This characterization of the relationship between direct and inverse theorems does not take into account the fact that the condition of the direct theorem is taken as given, without proof, so that its correctness is not guaranteed. The condition of the inverse theorem is not taken as given, since it is the conclusion of the proven direct theorem. Its correctness is confirmed by the proof of the direct theorem. This essential logical difference between the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and which cannot be proved by the logical method from the contrary.
Let's assume that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. We formulate it in a general form in a short form as follows: from A should E . Symbol A has the value of the given condition of the theorem, accepted without proof. Symbol E is the conclusion of the theorem to be proved.
We will prove the direct theorem by contradiction, logical method. The logical method proves a theorem that has not mathematical condition, and logical condition. It can be obtained if the mathematical condition of the theorem from A should E , supplement with the opposite condition from A do not do it E .
As a result, a logical contradictory condition of the new theorem was obtained, which includes two parts: from A should E And from A do not do it E . The resulting condition of the new theorem corresponds to the logical law of the excluded middle and corresponds to the proof of the theorem by contradiction.
According to the law, one part of the contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has its own task and goal to establish exactly which part of the two parts of the condition of the theorem is false. As soon as the false part of the condition is determined, it will be established that the other part is the true part, and the third is excluded.
According to the explanatory dictionary of mathematical terms, “proof is reasoning, during which the truth or falsity of any statement (judgment, statement, theorem) is established”. Proof contrary there is a discussion in the course of which it is established falsity(absurdity) of the conclusion that follows from false conditions of the theorem being proved.
Given: from A should E and from A do not do it E .
Prove: from A should E .
Proof: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its resolution in the proof and its result. The result turns out to be false if the reasoning is flawless and infallible. The reason for a false conclusion with logically correct reasoning can only be a contradictory condition: from A should E And from A do not do it E .
There is no shadow of a doubt that one part of the condition is false, and the other in this case is true. Both parts of the condition have the same origin, are accepted as given, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature has been found that would distinguish one part of the condition from the other. Therefore, to the same extent, from A should E and maybe from A do not do it E . Statement from A should E May be false, then the statement from A do not do it E will be true. Statement from A do not do it E may be false, then the statement from A should E will be true.
Therefore, it is impossible to prove the direct theorem by contradiction method.
Now we will prove the same direct theorem by the usual mathematical method.
Given: A .
Prove: from A should E .
Proof.
1. From A should B
2. From B should IN (according to the previously proved theorem)).
3. From IN should G (according to the previously proved theorem).
4. From G should D (according to the previously proved theorem).
5. From D should E (according to the previously proved theorem).
Based on the law of transitivity, from A should E . The direct theorem is proved by the usual method.
Let the proven direct theorem have a correct converse theorem: from E should A .
Let's prove it by ordinary mathematical method. The proof of the inverse theorem can be expressed in symbolic form as an algorithm of mathematical operations.
Given: E
Prove: from E should A .
Proof.
1. From E should D
2. From D should G (by the previously proved inverse theorem).
3. From G should IN (by the previously proved inverse theorem).
4. From IN do not do it B (the converse is not true). That's why from B do not do it A .
In this situation, it makes no sense to continue the mathematical proof of the inverse theorem. The reason for the situation is logical. It is impossible to replace an incorrect inverse theorem with anything. Therefore, this inverse theorem cannot be proved by the usual mathematical method. All hope is to prove this inverse theorem by contradiction.
In order to prove it by contradiction, it is required to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.
Inverse theorem claims: from E do not do it A . Her condition E , from which follows the conclusion A , is the result of proving the direct theorem by the usual mathematical method. This condition must be retained and supplemented with the statement from E should A . As a result of the addition, a contradictory condition of the new inverse theorem is obtained: from E should A And from E do not do it A . Based on this logically contradictory condition, the converse theorem can be proved by the correct logical reasoning only, and only, logical opposite method. In a proof by contradiction, any mathematical actions and operations are subordinate to logical ones and therefore do not count.
In the first part of the contradictory statement from E should A condition E was proved by the proof of the direct theorem. In the second part from E do not do it A condition E was assumed and accepted without proof. One of them is false and the other is true. It is required to prove which of them is false.
We prove with the correct logical reasoning and find that its result is a false, absurd conclusion. The reason for a false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. The false part can only be a statement from E do not do it A , in which E accepted without proof. This is what distinguishes it from E statements from E should A , which is proved by the proof of the direct theorem.
Therefore, the statement is true: from E should A , which was to be proved.
Conclusion: only that converse theorem is proved by the logical method from the contrary, which has a direct theorem proved by the mathematical method and which cannot be proved by the mathematical method.
The conclusion obtained acquires an exceptional importance in relation to the method of proof by contradiction of Fermat's great theorem. The overwhelming majority of attempts to prove it are based not on the usual mathematical method, but on the logical method of proving by contradiction. The proof of Fermat Wiles' Great Theorem is no exception.
Dmitry Abrarov in the article "Fermat's Theorem: the Phenomenon of Wiles' Proofs" published a commentary on the proof of Fermat's Last Theorem by Wiles. According to Abrarov, Wiles proves Fermat's Last Theorem with the help of a remarkable finding by the German mathematician Gerhard Frey (b. 1944) relating a potential solution to Fermat's equation x n + y n = z n
, Where n > 2
, with another completely different equation. This new equation is given by a special curve (called the Frey elliptic curve). The Frey curve is given by a very simple equation:
.
“It was precisely Frey who compared to every solution (a, b, c) Fermat's equation, that is, numbers satisfying the relation a n + b n = c n the above curve. In this case, Fermat's Last Theorem would follow."(Quote from: Abrarov D. "Fermat's Theorem: the phenomenon of Wiles proof")
In other words, Gerhard Frey suggested that the equation of Fermat's Last Theorem x n + y n = z n
, Where n > 2
, has solutions in positive integers. The same solutions are, by Frey's assumption, the solutions of his equation
y 2 + x (x - a n) (y + b n) = 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable discovery of Frey and, with its help, through mathematical method proved that this finding, that is, Frey's elliptic curve, does not exist. Therefore, there is no equation and its solutions that are given by a non-existent elliptic curve. Therefore, Wiles should have concluded that there is no equation of Fermat's Last Theorem and Fermat's Theorem itself. However, he takes the more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.
It may be an undeniable fact that Wiles accepted an assumption that is directly opposite in meaning to what is stated by Fermat's Last Theorem. It obliges Wiles to prove Fermat's Last Theorem by contradiction. Let's follow his example and see what happens from this example.
Fermat's Last Theorem states that the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers.
According to the logical method of proof by contradiction, this statement is preserved, accepted as given without proof, and then supplemented with a statement opposite in meaning: the equation x n + y n = z n , Where n > 2 , has solutions in positive integers.
The hypothesized statement is also accepted as given, without proof. Both statements, considered from the point of view of the basic laws of logic, are equally admissible, equal in rights and equally possible. By correct reasoning, it is required to establish which of them is false, in order to then establish that the other statement is true.
Correct reasoning ends with a false, absurd conclusion, the logical cause of which can only be a contradictory condition of the theorem being proved, which contains two parts of a directly opposite meaning. They were the logical cause of the absurd conclusion, the result of proof by contradiction.
However, in the course of logically correct reasoning, not a single sign was found by which it would be possible to establish which particular statement is false. It can be a statement: the equation x n + y n = z n , Where n > 2 , has solutions in positive integers. On the same basis, it can be the statement: the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers.
As a result of the reasoning, there can be only one conclusion: Fermat's Last Theorem cannot be proven by contradiction.
It would be a very different matter if Fermat's Last Theorem were an inverse theorem that has a direct theorem proved by the usual mathematical method. In this case, it could be proven by contradiction. And since it is a direct theorem, its proof must be based not on the logical method of proof by contradiction, but on the usual mathematical method.
According to D. Abrarov, Academician V. I. Arnold, the most famous contemporary Russian mathematician, reacted to Wiles's proof "actively skeptical". The academician stated: “this is not real mathematics – real mathematics is geometric and has strong links with physics.”
By contradiction, it is impossible to prove either that the equation of Fermat's Last Theorem has no solutions, or that it has solutions. Wiles' mistake is not mathematical, but logical - the use of proof by contradiction where its use does not make sense and does not prove Fermat's Last Theorem.
Fermat's Last Theorem is not proved with the help of the usual mathematical method, if it is given: the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers, and if it is required to prove in it: the equation x n + y n = z n , Where n > 2 , has no solutions in positive integers. In this form, there is not a theorem, but a tautology devoid of meaning.
Note. My BTF proof was discussed on one of the forums. One of the participants in Trotil, a specialist in number theory, made the following authoritative statement entitled: "A brief retelling of what Mirgorodsky did." I quote it verbatim:
« A. He proved that if z 2 \u003d x 2 + y , That z n > x n + y n . This is a well-known and quite obvious fact.
IN. He took two triples - Pythagorean and non-Pythagorean and showed by simple enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is performed (and only for it).
WITH. And then the author omitted the fact that from < in a subsequent degree may be = , not only > . A simple counterexample is the transition n=1 V n=2 in a Pythagorean triple.
D. This point does not contribute anything essential to the BTF proof. Conclusion: BTF has not been proven.”
I will consider his conclusion point by point.
A. In it, the BTF is proved for the entire infinite set of triples of Pythagorean numbers. Proven by a geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was opened, as I believe, by P. Fermat himself. Fermat might have had this in mind when he wrote:
"I have discovered a truly marvelous proof of this, but these margins are too narrow for it." This assumption of mine is based on the fact that in the Diophantine problem, against which, in the margins of the book, Fermat wrote, we are talking about solutions to the Diophantine equation, which are triples of Pythagorean numbers.
An infinite set of triples of Pythagorean numbers are solutions to the Diophatian equation, and in Fermat's theorem, on the contrary, none of the solutions can be a solution to the equation of Fermat's theorem. And Fermat's truly miraculous proof has a direct bearing on this fact. Later, Fermat could extend his theorem to the set of all natural numbers. On the set of all natural numbers, BTF does not belong to the "set of exceptionally beautiful theorems". This is my assumption, which can neither be proved nor disproved. It can be both accepted and rejected.
IN. In this paragraph, I prove that both the family of an arbitrarily taken Pythagorean triple of numbers and the family of an arbitrarily taken non-Pythagorean triple of numbers BTF is satisfied. This is a necessary, but insufficient and intermediate link in my proof of the BTF. The examples I have taken of the family of a triple of Pythagorean numbers and the family of a triple of non-Pythagorean numbers have the meaning of specific examples that presuppose and do not exclude the existence of similar other examples.
Trotil's statement that I "showed by simple enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is fulfilled (and only for it) is without foundation. He cannot refute the fact that I could just as well take other examples of Pythagorean and non-Pythagorean triples to get a specific family of one and the other triple.
Whatever pair of triples I take, checking their suitability for solving the problem can be carried out, in my opinion, only by the method of "simple enumeration". Any other method is not known to me and is not required. If he did not like Trotil, then he should have suggested another method, which he does not. Without offering anything in return, it is incorrect to condemn “simple enumeration”, which in this case is irreplaceable.
WITH. I omitted = between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 \u003d x 2 + y (1), in which the degree n > 2 — whole positive number. From the equality between the inequalities it follows obligatory consideration of equation (1) with a non-integer value of the degree n > 2 . Trotil counting compulsory consideration of equality between inequalities, actually considers necessary in the BTF proof, consideration of equation (1) with non-integer degree value n > 2 . I did this for myself and found that equation (1) with non-integer degree value n > 2 has a solution of three numbers: z, (z-1), (z-1) with a non-integer exponent.
For integers n greater than 2, the equation x n + y n = z n has no non-zero solutions in natural numbers.
You probably remember from your school days the Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. You may also remember the classic right triangle with sides whose lengths are related as 3: 4: 5. For it, the Pythagorean theorem looks like this:
This is an example of solving the generalized Pythagorean equation in non-zero integers for n= 2. Fermat's Last Theorem (also called "Fermat's Last Theorem" and "Fermat's Last Theorem") is the statement that, for values n> 2 equations of the form x n + y n = z n do not have nonzero solutions in natural numbers.
The history of Fermat's Last Theorem is very entertaining and instructive, and not only for mathematicians. Pierre de Fermat contributed to the development of various areas of mathematics, but the main part of his scientific heritage was published only posthumously. The fact is that mathematics for Fermat was something like a hobby, not a professional occupation. He corresponded with the leading mathematicians of his time, but did not seek to publish his work. Fermat's scientific writings are mostly found in the form of private correspondence and fragmentary notes, often made in the margins of various books. It is on the margins (of the second volume of the ancient Greek Arithmetic by Diophantus. - Note. translator) shortly after the death of the mathematician, the descendants discovered the formulation of the famous theorem and the postscript:
« I found a truly wonderful proof of this, but these margins are too narrow for him.».
Alas, apparently, Fermat never bothered to write down the “miraculous proof” he found, and descendants unsuccessfully searched for it for more than three centuries. Of all Fermat's disparate scientific heritage, containing many surprising statements, it was the Great Theorem that stubbornly resisted solution.
Whoever did not take up the proof of Fermat's Last Theorem - all in vain! Another great French mathematician, René Descartes (René Descartes, 1596-1650), called Fermat a "braggart", and the English mathematician John Wallis (John Wallis, 1616-1703) called him a "damn Frenchman". Fermat himself, however, nevertheless left behind a proof of his theorem for the case n= 4. With proof for n= 3 was solved by the great Swiss-Russian mathematician of the 18th century Leonard Euler (1707–83), after which, having failed to find proofs for n> 4, jokingly offered to search Fermat's house to find the key to the lost evidence. In the 19th century, new methods of number theory made it possible to prove the statement for many integers within 200, but, again, not for all.
In 1908 a prize of DM 100,000 was established for this task. The prize fund was bequeathed to the German industrialist Paul Wolfskehl, who, according to legend, was about to commit suicide, but was so carried away by Fermat's Last Theorem that he changed his mind about dying. With the advent of adding machines, and then computers, the bar of values n began to rise higher and higher - up to 617 by the beginning of World War II, up to 4001 in 1954, up to 125,000 in 1976. At the end of the 20th century, the most powerful computers of military laboratories in Los Alamos (New Mexico, USA) were programmed to solve the Fermat problem in the background (similar to the screen saver mode of a personal computer). Thus, it was possible to show that the theorem is true for incredibly large values x, y, z And n, but this could not serve as a rigorous proof, since any of the following values n or triples of natural numbers could disprove the theorem as a whole.
Finally, in 1994, the English mathematician Andrew John Wiles (Andrew John Wiles, b. 1953), while working at Princeton, published a proof of Fermat's Last Theorem, which, after some modifications, was considered exhaustive. The proof took more than a hundred magazine pages and was based on the use of the modern apparatus of higher mathematics, which had not been developed in Fermat's era. So what, then, did Fermat mean by leaving a message in the margins of the book that he had found proof? Most of the mathematicians I have spoken to on this subject have pointed out that over the centuries there have been more than enough incorrect proofs of Fermat's Last Theorem, and that it is likely that Fermat himself found a similar proof but failed to see the error in it. However, it is possible that there is still some short and elegant proof of Fermat's Last Theorem, which no one has yet found. Only one thing can be said with certainty: today we know for sure that the theorem is true. Most mathematicians, I think, would unreservedly agree with Andrew Wiles, who remarked about his proof, "Now at last my mind is at peace."
Many years ago, I received a letter from Tashkent from Valery Muratov, judging by the handwriting, a man of youthful age, who then lived on Kommunisticheskaya Street in the house number 31. The guy was determined: “Directly to the point. How much will you pay me for proving Fermat’s theorem? suits at least 500 rubles. At another time, I would have proved it to you for free, but now I need money ... "
An amazing paradox: few people know who Fermat is, when he lived and what he did. Even fewer people can even describe his great theorem in the most general terms. But everyone knows that there is some kind of Fermat's theorem, over the proof of which mathematicians of the whole world have been struggling for more than 300 years, but they cannot prove it!
There are many ambitious people, and the very consciousness that there is something that others cannot do, further spurs their ambition. Therefore, thousands (!) of proofs of the Great Theorem have come and come to academies, scientific institutes, and even newspaper editorial offices around the world - an unprecedented and never broken record of pseudoscientific amateur performance. There is even a term: "fermatists", that is, people obsessed with the desire to prove the Great Theorem, who completely exhausted professional mathematicians with demands to evaluate their work. The famous German mathematician Edmund Landau even prepared a standard, according to which he answered: "There is an error on the page in your proof of Fermat's theorem ...", and his graduate students put down the page number. And in the summer of 1994, newspapers around the world report something completely sensational: The Great Theorem is proved!
So, who is Fermat, what is the essence of the problem and has it really been solved? Pierre Fermat was born in 1601 in the family of a tanner, a wealthy and respected man - he served as second consul in his native town of Beaumont - this is something like an assistant to the mayor. Pierre studied first with the Franciscan monks, then at the Faculty of Law in Toulouse, where he then practiced advocacy. However, Fermat's range of interests went far beyond jurisprudence. He was especially interested in classical philology, his comments on the texts of ancient authors are known. And the second passion is mathematics.
In the 17th century, as, indeed, for many years later, there was no such profession: mathematician. Therefore, all the great mathematicians of that time were "part-time" mathematicians: Rene Descartes served in the army, Francois Viet was a lawyer, Francesco Cavalieri was a monk. There were no scientific journals then, and the classic of science Pierre Fermat did not publish a single scientific work during his lifetime. There was a rather narrow circle of "amateurs" who solved various interesting problems for them and wrote letters to each other about this, sometimes arguing (like Fermat with Descartes), but, basically, remained like-minded. They became the founders of new mathematics, the sowers of brilliant seeds, from which the mighty tree of modern mathematical knowledge began to grow, gaining strength and branching.
So, Fermat was the same "amateur". In Toulouse, where he lived for 34 years, everyone knew him, first of all, as an adviser to the Chamber of Investigation and an experienced lawyer. At the age of 30, he married, had three sons and two daughters, sometimes went on business trips, and during one of them he died suddenly at the age of 63. All! The life of this man, a contemporary of the Three Musketeers, is surprisingly uneventful and devoid of adventure. Adventures fell to the share of his Great Theorem. We will not talk about Fermat's entire mathematical heritage, and it is difficult to talk about him in a popular way. Take my word for it: this legacy is great and varied. The assertion that the Great Theorem is the pinnacle of his work is highly debatable. It's just that the fate of the Great Theorem is surprisingly interesting, and the vast world of people uninitiated in the mysteries of mathematics has always been interested not in the theorem itself, but in everything around it...
The roots of this whole story must be sought in antiquity, so beloved by Fermat. Approximately in the 3rd century, the Greek mathematician Diophantus lived in Alexandria, a scientist who thought in an original way, thinking outside the box and expressing his thoughts outside the box. Of the 13 volumes of his Arithmetic, only 6 have come down to us. Just when Fermat was 20 years old, a new translation of his works came out. Fermat was very fond of Diophantus, and these writings were his reference book. On its fields, Fermat wrote down his Great Theorem, which in its simplest modern form looks like this: the equation Xn + Yn = Zn has no solution in integers for n - more than 2. (For n = 2, the solution is obvious: Z2 + 42 = 52 ). In the same place, on the margins of the Diophantine volume, Fermat adds: "I discovered this truly wonderful proof, but these margins are too narrow for him."
At first glance, the little thing is simple, but when other mathematicians began to prove this "simple" theorem, no one succeeded for a hundred years. Finally, the great Leonhard Euler proved it for n = 4, then after 20 (!) years - for n = 3. And again the work stalled for many years. The next victory belongs to the German Peter Dirichlet (1805–1859) and the Frenchman Andrien Legendre (1752–1833), who admitted that Fermat was right for n = 5. Then the Frenchman Gabriel Lamet (1795–1870) did the same for n = 7. Finally, in the middle of the last century, the German Ernst Kummer (1810-1893) proved the Great Theorem for all values of n less than or equal to 100. Moreover, he proved it using methods that could not be known to Fermat, which further strengthened the veil of mystery around the Great Theorem.
Thus, it turned out that they were proving Fermat's theorem "piece by piece", but no one was able to "completely". New attempts at proofs only led to a quantitative increase in the values of n. Everyone understood that, having spent an abyss of labor, it was possible to prove the Great Theorem for an arbitrarily large number n, but Fermat spoke about any value of it greater than 2! It was in this difference between "arbitrarily large" and "any" that the whole meaning of the problem was concentrated.
However, it should be noted that attempts to prove Fermg's theorem were not just some kind of mathematical game, the solution of a complex rebus. In the course of these proofs, new mathematical horizons were opened up, problems arose and solved, which became new branches of the mathematical tree. The great German mathematician David Hilbert (1862-1943) cited the Great Theorem as an example of "what a stimulating effect a special and seemingly insignificant problem can have on science." The same Kummer, working on Fermat's theorem, himself proved theorems that formed the foundation of number theory, algebra and function theory. So proving the Great Theorem is not a sport, but a real science.
Time passed, and electronics came to the aid of professional "fsrmatnts". Electronic brains of new methods could not be invented, but they took speed. Around the beginning of the 80s, Fermat's theorem was proved with the help of a computer for n less than or equal to 5500. Gradually, this figure grew to 100,000, but everyone understood that such "accumulation" was a matter of pure technology, giving nothing to the mind or heart . They could not take the fortress of the Great Theorem "head on" and began to look for roundabout maneuvers.
In the mid-1980s, the young mathematician G. Filettings proved the so-called "Mordell's conjecture", which, by the way, was also "unreachable" by any of the mathematicians for 61 years. The hope arose that now, so to speak, "attacking from the flank", Fermat's theorem could also be solved. However, nothing happened then. In 1986, the German mathematician Gerhard Frei proposed a new proof method in Essesche. I do not undertake to explain it strictly, but not in mathematical, but in general human language, it sounds something like this: if we are convinced that the proof of some other theorem is an indirect, in some way transformed proof of Fermat's theorem, then, therefore, we will prove the Great Theorem. A year later, the American Kenneth Ribet from Berkeley showed that Frey was right and, indeed, one proof could be reduced to another. Many mathematicians around the world have taken this path. We have done a lot to prove the Great Theorem by Viktor Aleksandrovich Kolyvanov. The three-hundred-year-old walls of the impregnable fortress trembled. Mathematicians realized that it would not last long.
In the summer of 1993, in ancient Cambridge, at the Isaac Newton Institute of Mathematical Sciences, 75 of the world's most prominent mathematicians gathered to discuss their problems. Among them was the American professor Andrew Wiles of Princeton University, a prominent specialist in number theory. Everyone knew that he had been working on the Great Theorem for many years. Wiles made three presentations, and at the last one, on June 23, 1993, at the very end, turning away from the blackboard, he said with a smile:
I guess I won't continue...
There was dead silence at first, then a round of applause. Those sitting in the hall were qualified enough to understand: Fermat's Last Theorem is proved! In any case, none of those present found any errors in the above proof. Associate director of the Newton Institute, Peter Goddard, told reporters:
“Most experts didn't think they'd find out for the rest of their lives. This is one of the greatest achievements of mathematics of our century...
Several months have passed, no comments or denials followed. True, Wiles did not publish his proof, but only sent the so-called prints of his work to a very narrow circle of his colleagues, which, naturally, prevents mathematicians from commenting on this scientific sensation, and I understand Academician Ludwig Dmitrievich Faddeev, who said:
- I can say that the sensation happened when I see the proof with my own eyes.
Faddeev believes that the likelihood of Wiles winning is very high.
“My father, a well-known specialist in number theory, was, for example, sure that the theorem would be proved, but not by elementary means,” he added.
Another academician of ours, Viktor Pavlovich Maslov, was skeptical about the news, and he believes that the proof of the Great Theorem is not an actual mathematical problem at all. In terms of his scientific interests, Maslov, the chairman of the Council for Applied Mathematics, is far from "fermatists", and when he says that the complete solution of the Great Theorem is only of sporting interest, one can understand him. However, I dare to note that the concept of relevance in any science is a variable. 90 years ago, Rutherford, probably, was also told: "Well, well, well, the theory of radioactive decay ... So what? What is the use of it? .."
The work on the proof of the Great Theorem has already given a lot of mathematics, and one can hope that it will give more.
“What Wiles has done will move mathematicians into other areas,” said Peter Goddard. - Rather, this does not close one of the lines of thought, but raises new questions that will require an answer ...
Professor of Moscow State University Mikhail Ilyich Zelikin explained the current situation to me this way:
Nobody sees any mistakes in Wiles's work. But for this work to become a scientific fact, it is necessary that several reputable mathematicians independently repeat this proof and confirm its correctness. This is an indispensable condition for the recognition of Wiles' work by the mathematical community...
How long will it take for this?
I asked this question to one of our leading specialists in the field of number theory, Doctor of Physical and Mathematical Sciences Alexei Nikolaevich Parshin.
Andrew Wiles has a lot of time ahead of him...
The fact is that on September 13, 1907, the German mathematician P. Wolfskel, who, unlike the vast majority of mathematicians, was a rich man, bequeathed 100 thousand marks to the one who would prove the Great Theorem in the next 100 years. At the beginning of the century, interest from the bequeathed amount went to the treasury of the famous Getgangent University. This money was used to invite leading mathematicians to give lectures and conduct scientific work. At that time, David Hilbert, whom I have already mentioned, was chairman of the award commission. He did not want to pay the premium.
“Fortunately,” said the great mathematician, “it seems that we don’t have a mathematician, except for me, who would be able to do this task, but I will never dare to kill the goose that lays golden eggs for us.”
Before the deadline - 2007, designated by Wolfskel, there are few years left, and, it seems to me, a serious danger looms over "Hilbert's chicken". But it's not about the prize, actually. It's about the inquisitiveness of thought and human perseverance. They fought for more than three hundred years, but they still proved it!
And further. For me, the most interesting thing in this whole story is: how did Fermat himself prove his Great Theorem? After all, all today's mathematical tricks were unknown to him. And did he prove it at all? After all, there is a version that he seemed to have proved, but he himself found an error, and therefore he did not send the proofs to other mathematicians, but forgot to cross out the entry in the margins of the Diophantine volume. Therefore, it seems to me that the proof of the Great Theorem, obviously, took place, but the secret of Fermat's theorem remained, and it is unlikely that we will ever reveal it ...
Perhaps Fermat was mistaken then, but he was not mistaken when he wrote: “Perhaps posterity will be grateful to me for showing him that the ancients did not know everything, and this may penetrate the consciousness of those who will come after me. to pass the torch to his sons..."
There are not so many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only mathematical problem that has become so widely known and has become a real legend. It is mentioned in many books and films, while the main context of almost all mentions is the impossibility of proving the theorem.
Yes, this theorem is very famous and in a sense has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this happened back in 1995. But first things first.
So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in nature and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n> 2. Everything seems to be simple and clear, but the best mathematicians and simple amateurs fought over searching for a solution for more than three and a half centuries.
Why is she so famous? Now let's find out...
Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with 5 grades of secondary school, but the proof is far from even every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved for so long. 2. What does it consist of?
Let's start with Pythagorean pants The wording is really simple - at first glance. As we know from childhood, "Pythagorean pants are equal on all sides." The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs.
In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triples satisfying the equation x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They probably tried to look for triples and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.
That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²
Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.
It is more difficult to prove the absence: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give a solution). And that's it, the opponent is defeated. How to prove absence?
To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.
In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2): 
And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:
But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically studied the general equation x n + y n \u003d z n. And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! All that remains is his remark in Diophantus' Arithmetic: "I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it."
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.
After Fermat, such great minds as Leonhard Euler worked on the search for proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat's Last Theorem, but only in 1993 did mathematicians see and believe that the three-century saga of finding a proof of Fermat's last theorem was almost over.
It is easy to show that it suffices to prove Fermat's theorem only for prime n: 3, 5, 7, 11, 13, 17, … For composite n, the proof remains valid. But there are infinitely many prime numbers...
In 1825, using the method of Sophie Germain, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.
Finally, the German mathematician Ernst Kummer showed in a brilliant study that the methods of mathematics in the 19th century cannot prove the theorem in general form. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.
In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.
He soon died of natural causes. The heirs were pretty surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:
Dear (s). . . . . . . .
Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, drawing on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems, the continuum hypothesis. What if Fermat's Last Theorem is also unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups of programmers and mathematicians proved Fermat's Last Theorem for all values of n up to 500, then up to 1,000, and later up to 10,000.
In the 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians claimed that Fermat's Last Theorem was true for all values of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.
Nevertheless, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole building could collapse at any moment.
In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama-Shimura hypothesis. Having proved that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proved. But for thirty years it was not possible to prove the Taniyama-Shimura hypothesis, and there were less and less hopes for success.
In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.
Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama-Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat’s Last Theorem is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.
In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.
While the hype continued in the press, serious work began to verify the evidence. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.
It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.
“...half a minute after the start of the festive dinner on the occasion of her birthday, I gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?
This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ...
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