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 agree wholeheartedly 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 came and continue to 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 different 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 margins, 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 at 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 a lot of work, it is 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 process 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, a young nemeadny mathematician G. Filitings 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 don’t undertake to explain it strictly, but not in mathematical, but in general human language, it sounds something like this: if we make sure 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..."
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 power 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 x 2 + y 2 = z 2
(1) has an infinite set of solutions on the set of all triples of Pythagorean numbers, then, conversely, the equation x n + y n = z n
, where n > 2
(2) has no solutions on the set of positive integers."
Proof.
BUT) Everyone knows that equation (1) has an infinite number of solutions on the set of all triples of Pythagorean numbers. Let us prove that no triple of Pythagorean numbers, which is a solution to equation (1), is a solution to equation (2).
Based on the law of reversibility of equality, the sides of equation (1) are interchanged. Pythagorean numbers (z, x, y) can be interpreted as the lengths of the sides of a right triangle, and the squares (x2, y2, z2) can be interpreted as the areas of squares built on its hypotenuse and legs.
We multiply the squares of equation (1) by an arbitrary height h :
z 2 h = x 2 h + y 2 h (3)
Equation (3) can be interpreted as the equality of the volume of a parallelepiped to the sum of the volumes of two parallelepipeds.
Let the height of three parallelepipeds h = z :
z 3 = x 2 z + y 2 z (4)
The volume of the cube is decomposed into two volumes of two parallelepipeds. We leave the volume of the cube unchanged, and reduce the height of the first parallelepiped to x and the height of the second parallelepiped will be reduced to y . The volume of a cube is greater than the sum of the volumes of two cubes:
z 3 > x 3 + y 3 (5)
On the set of triples of Pythagorean numbers ( x, y, z ) at n=3 there can be no solution to equation (2). Consequently, on the set of all triples of Pythagorean numbers, it is impossible to decompose a cube into two cubes.
Let in equation (3) the height of three parallelepipeds h = z2 :
z 2 z 2 = x 2 z 2 + y 2 z 2 (6)
The volume of a parallelepiped is decomposed into the sum of the volumes of two parallelepipeds.
We leave the left side of equation (6) unchanged. On its right side the height z2
reduce to X
in the first term and up to at 2
in the second term.
Equation (6) turned into the inequality:
The volume of a parallelepiped is decomposed into two volumes of two parallelepipeds.
We leave the left side of equation (8) unchanged.
On the right side of the height zn-2
reduce to xn-2
in the first term and reduce to y n-2
in the second term. Equation (8) turns into the inequality:
| 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.
FROM) Fermat's commentary on the Diophantus problem states that it is impossible to decompose " in general, no power greater than the square, two powers with the same exponent».
Kisses a power greater than a square cannot really be decomposed into two powers with the same exponent. I don't kiss a power greater than the square can be decomposed into two powers with the same exponent.
Any randomly chosen triple of positive integers (z, x, y) may belong to a family, each member of which consists of a constant number z and two numbers less than z . Each member of the family can be represented in the form of an inequality, and all the resulting inequalities can be represented as a sequence of inequalities:
| 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 of the left side of the sequence change their meaning and take on the meaning of the inequalities of the right side of the inequalities of the sequence (14). As a result of the increase in the exponent of all inequalities, the left side is greater than the right side:
| 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 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 declare openly that the method of proof by contradiction is not a mathematical method, although it is used in mathematics, that it is a logical method and belongs to logic. Is it valid to say that proof by contradiction is "used whenever a direct theorem is difficult to prove", when in fact it is used if, and only if, there is no substitute for it.
The characteristic of the relationship between the direct and inverse theorems also deserves special attention. “An inverse theorem for a given theorem (or to a given theorem) is a theorem in which the condition is the conclusion, and the conclusion is the condition of the given theorem. This theorem in relation to the converse theorem is called the direct theorem (initial). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and inverse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If the diagonals in a quadrilateral are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e., the converse theorem is not true./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.261 /.
This characterization of the relationship between direct and inverse theorems does not take into account the fact that the condition of the direct theorem is taken as given, without proof, so that its correctness is not guaranteed. The condition of the inverse theorem is not taken as given, since it is the conclusion of the proven direct theorem. Its correctness is confirmed by the proof of the direct theorem. This essential logical difference between the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and which cannot be proved by the logical method from the contrary.
Let's assume that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. We formulate it in a general form in a short form as follows: from BUT should E . Symbol BUT has the value of the given condition of the theorem, accepted without proof. Symbol E is the conclusion of the theorem to be proved.
We will prove the direct theorem by contradiction, logical method. The logical method proves a theorem that has not mathematical condition, and logical condition. It can be obtained if the mathematical condition of the theorem from BUT should E , supplement with the opposite condition from BUT it does not follow E .
As a result, a logical contradictory condition of the new theorem was obtained, which includes two parts: from BUT should E and from BUT 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 the contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has its own task and goal to establish exactly which part of the two parts of the condition of the theorem is false. As soon as the false part of the condition is determined, it will be established that the other part is the true part, and the third is excluded.
According to the explanatory dictionary of mathematical terms, “proof is reasoning, during which the truth or falsity of any statement (judgment, statement, theorem) is established”. Proof contrary there is a discussion in the course of which it is established falsity(absurdity) of the conclusion that follows from false conditions of the theorem being proved.
Given: from BUT should E and from BUT it does not follow E .
Prove: from BUT should E .
Proof: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its resolution in the proof and its result. The result turns out to be false if the reasoning is flawless and infallible. The reason for a false conclusion with logically correct reasoning can only be a contradictory condition: from BUT should E and from BUT 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 given, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature has been found that would distinguish one part of the condition from the other. Therefore, to the same extent, from BUT should E and maybe from BUT it does not follow E . Statement from BUT should E may be false, then the statement from BUT it does not follow E will be true. Statement from BUT it does not follow E may be false, then the statement from BUT should E will be true.
Therefore, it is impossible to prove the direct theorem by contradiction method.
Now we will prove the same direct theorem by the usual mathematical method.
Given: BUT .
Prove: from BUT should E .
Proof.
1. From BUT should B
2. From B should AT (according to the previously proved theorem)).
3. From AT should G (according to the previously proved theorem).
4. From G should D (according to the previously proved theorem).
5. From D should E (according to the previously proved theorem).
Based on the law of transitivity, from BUT should E . The direct theorem is proved by the usual method.
Let the proven direct theorem have a correct converse theorem: from E should BUT .
Let's prove it by ordinary mathematical method. The proof of the inverse theorem can be expressed in symbolic form as an algorithm of mathematical operations.
Given: E
Prove: from E should BUT .
Proof.
1. From E should D
2. From D should G (by the previously proved inverse theorem).
3. From G should AT (by the previously proved inverse theorem).
4. From AT it does not follow B (the converse is not true). That's why from B it does not follow BUT .
In this situation, it makes no sense to continue the mathematical proof of the inverse theorem. The reason for the situation is logical. It is impossible to replace an incorrect inverse theorem with anything. Therefore, this inverse theorem cannot be proved by the usual mathematical method. All hope is to prove this inverse theorem by contradiction.
In order to prove it by contradiction, it is required to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.
Inverse theorem claims: from E it does not follow BUT . Her condition E , from which follows the conclusion BUT , is the result of proving the direct theorem by the usual mathematical method. This condition must be retained and supplemented with the statement from E should BUT . As a result of the addition, a contradictory condition of the new inverse theorem is obtained: from E should BUT and from E it does not follow BUT . Based on this logically contradictory condition, the converse theorem can be proved by the correct logical reasoning only, and only, logical opposite method. In a proof by contradiction, any mathematical actions and operations are subordinate to logical ones and therefore do not count.
In the first part of the contradictory statement from E should BUT condition E was proved by the proof of the direct theorem. In the second part from E it does not follow BUT condition E was assumed and accepted without proof. One of them is false and the other is true. It is required to prove which of them is false.
We prove with the correct logical reasoning and find that its result is a false, absurd conclusion. The reason for a false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. The false part can only be a statement from E it does not follow BUT , wherein E accepted without proof. This is what distinguishes it from E statements from E should BUT , which is proved by the proof of the direct theorem.
Therefore, the statement is true: from E should BUT , which was to be proved.
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 his article "Fermat's Theorem: the Phenomenon of Wiles' Proofs" published a commentary on the proof of Fermat's Last Theorem by Wiles. According to Abrarov, Wiles proves Fermat's Last Theorem with the help of a remarkable finding by the German mathematician Gerhard Frey (b. 1944) relating a potential solution to Fermat's equation x n + y n = z n
, where n > 2
, with another completely different equation. This new equation is given by a special curve (called the Frey elliptic curve). The Frey curve is given by a very simple equation:
.
“It was precisely Frey who compared to every solution (a, b, c) Fermat's equation, that is, numbers satisfying the relation a n + b n = c n the above curve. In this case, Fermat's Last Theorem would follow."(Quote from: Abrarov D. "Fermat's Theorem: the phenomenon of Wiles proof")
In other words, Gerhard Frey suggested that the equation of Fermat's Last Theorem x n + y n = z n
, where n > 2
, has solutions in positive integers. The same solutions are, by Frey's assumption, the solutions of his equation
y 2 + x (x - a n) (y + b n) = 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable discovery of Frey and, with its help, through mathematical method proved that this finding, that is, Frey's elliptic curve, does not exist. Therefore, there is no equation and its solutions that are given by a non-existent elliptic curve. Therefore, Wiles should have concluded that there is no equation of Fermat's Last Theorem and Fermat's Theorem itself. However, he takes the more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.
It may be an undeniable fact that Wiles accepted an assumption that is directly opposite in meaning to that which 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 exactly 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:
« BUT. He proved that if z 2 \u003d x 2 + y , then z n > x n + y n . This is a well-known and quite obvious fact.
AT. He took two triples - Pythagorean and non-Pythagorean and showed by simple enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is performed (and only for it).
FROM. And then the author omitted the fact that from < in a subsequent degree may be = , not only > . A simple counterexample is the transition n=1 in n=2 in a Pythagorean triple.
D. This point does not contribute anything essential to the BTF proof. Conclusion: BTF has not been proven.”
I will consider his conclusion point by point.
BUT. In it, the BTF is proved for the entire infinite set of triples of Pythagorean numbers. Proven by a geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was opened, as I believe, by P. Fermat himself. Fermat might have had this in mind when he wrote:
"I have discovered a truly marvelous proof of this, but these margins are too narrow for it." This assumption of mine is based on the fact that in the Diophantine problem, against which, in the margins of the book, Fermat wrote, we are talking about solutions to the Diophantine equation, which are triples of Pythagorean numbers.
An infinite set of triples of Pythagorean numbers are solutions to the Diophate 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.
AT. 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, certain family of triples (78 and 210 pieces) BTF is fulfilled (and only for it) is without foundation. He cannot refute the fact that I could just as well take other examples of Pythagorean and non-Pythagorean triples to get a specific family of one and the other triple.
Whatever pair of triples I take, checking their suitability for solving the problem can be carried out, in my opinion, only by the method of "simple enumeration". Any other method is not known to me and is not required. If he did not like Trotil, then he should have suggested another method, which he does not. Without offering anything in return, it is incorrect to condemn “simple enumeration”, which in this case is irreplaceable.
FROM. I omitted = between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 \u003d x 2 + y (1), in which the degree n > 2 — whole positive number. From the equality between the inequalities it follows obligatory consideration of equation (1) with a non-integer value of the degree n > 2 . Trotil counting compulsory consideration of equality between inequalities, actually considers necessary in the BTF proof, consideration of equation (1) with non-integer degree value n > 2 . I did this for myself and found that equation (1) with non-integer degree value n > 2 has a solution of three numbers: z, (z-1), (z-1) with a non-integer exponent.
HISTORY OF FERMAT'S GREAT 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 20th century there was one grandiose 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 by how keenly 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 speaker (and all of us, when we say speakers) is simply obliged to know the history of the theorem.
If it so happened that you do not like mathematics as much as I love it, then look at some deepenings in detail with a cursory glance. Understanding that not all readers of our mailing list are interested in wandering in the wilds of mathematics, I tried not to give any formulas (except for the equation of Fermat's theorem and a couple of hypotheses) and to simplify the coverage of some specific issues as much as possible.
How Fermat brewed porridge
The French lawyer and part-time 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 Fermat's Great (or Great) Theorem. This is one of the most famous and phenomenal mathematical theorems. Probably, the excitement around it would not have been so strong if in the book of Diophantus of Alexandria (3rd century AD) "Arithmetic", which Fermat often studied, making notes on its wide margins, and which his son Samuel kindly preserved for posterity , approximately the following entry of the great mathematician was not found:
"I have a very startling piece of evidence, but it's too big to fit in the margins."
It was this entry that caused the subsequent grandiose turmoil around the theorem.
So, the famous scientist said that he had proved his theorem. Let's ask ourselves the question: did he really prove it or did he lie corny? Or are there other versions explaining the appearance of that marginal entry that did not allow many mathematicians of the next generations to sleep peacefully?
The history of the Great Theorem is as fascinating as an adventure through time. Fermat stated in 1636 that an equation of the form x n + y n =z n has no solutions in integers with exponent n>2. This is actually Fermat's Last Theorem. In this seemingly simple mathematical formula, the universe has masked incredible complexity. The Scottish-born American mathematician Eric Temple Bell, in his book The Final Problem (1961), even suggested that perhaps humanity would cease to exist before it could prove Fermat's Last Theorem.
It is somewhat strange that for some reason the theorem was late with its birth, since the situation was long overdue, because its special case for n = 2 - another famous mathematical formula - the Pythagorean theorem, arose twenty-two centuries earlier. Unlike Fermat's theorem, the Pythagorean theorem has an infinite number 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) …
Grand Theorem Syndrome
Who just did not try to prove Fermat's theorem. Any fledgling student considered it his duty to apply to the Great Theorem, but no one was able to prove it. At first it didn't work for a hundred years. Then a hundred more. And further. A mass syndrome began to develop among mathematicians: "How is it? Fermat proved it, but what if I can't, or what?" - and some of them went crazy on this basis in the full sense of the word.
No matter how much the theorem was tested, it always turned out to be true. I knew one energetic programmer who was obsessed with the idea of disproving the Great Theorem by trying to find at least one of its solutions (counterexample) by iterating over integers using a fast computer (at that time more commonly called a computer). He believed in the success of his enterprise and liked 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 this kind of bold seekers. Of course, he did not find any solution. And no computers, even with fabulous speed, could ever test the theorem, because all the variables of this equation (including the exponents) can increase to infinity.
Theorem requires proof
Mathematicians know that if a theorem is not proven, anything (either true or false) can follow from it, as it did 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 (that is, they do not have integer divisors and are divisible only by themselves and by one without a remainder), if n is a power of two (1, 2, 4, 8, 16, 32, 64, etc.). Fermat's hypothesis lived for more than a hundred years - until Leonhard Euler showed in 1732 that
2 32 +1 = 4 294 967 297 = 6 700 417 641
Then, almost 150 years later (1880), Fortune Landry factored the following Fermat number:
2 64 +1 = 18 446 744 073 709 551 617 = 274 177 67 280 421 310 721
How they could find the divisors of these large numbers without the help of computers - God only knows. In turn, Euler put forward the hypothesis that the equation x 4 + y 4 + z 4 =u 4 has no solutions in integers. However, about 250 years later, in 1988, Naum Elkis from Harvard managed to discover (already using 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 proof, otherwise it was just a hypothesis, and it could well be that somewhere in the endless numerical fields the solution to the equation of the Great Theorem was lost.
The most virtuoso and prolific mathematician of the 18th century, Leonhard Euler, whose archive of records mankind has been sorting out for almost a century, proved Fermat's theorem for powers 3 and 4 (or rather, he repeated the lost proofs of Pierre Fermat himself); his follower in number theory, Legendre (and independently Dirichlet) - for degree 5; Lame - for degree 7. But in general terms, the theorem remained unproved.
On March 1, 1847, at a meeting of the Paris Academy of Sciences, two outstanding mathematicians at once - Gabriel Lame and Augustin Cauchy - announced that they had come to the end of the proof of the Great Theorem and arranged a race, publishing their proofs in parts. However, the duel between them was interrupted because the same error was discovered 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 Wolfskell's testament by the Göttingen Academy of Sciences, it was inundated with thousands of proofs from lovers of mathematics, and this flow did not stop for decades, but, as you can imagine, they all contained errors. They say that the academy prepared forms with the following content:
Dear __________________________!
In your proof of Fermat's Theorem on ____ page ____ line from the top
The following error was found in the formula:__________________________:,
Which were sent to unlucky applicants for the award.
At that time, a semi-contemptuous nickname appeared in the circle of mathematicians - fermist. This was the name given to any self-confident upstart who lacked knowledge, but more than had ambition to hastily try his hand at proving the Great Theorem, and then, not noticing his own mistakes, proudly slapping his chest, loudly declare: "I proved the first Fermat's Theorem! Every farmer, even if he was ten thousandth in number, considered himself the first - this was ridiculous. The simple appearance of the Great Theorem reminded Fermists of easy prey so much that they were not at all embarrassed that even Euler and Gauss could not cope with it.
(Fermists, oddly enough, still exist today. Although one of them did not believe that he had proved the theorem like a classical fermist, but until recently he 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 cautiously approach this unbearable rod, but did not talk about it aloud, so as not to be branded as Fermists and, thus, not to harm their high authority.
By that time, the proof of the theorem for the exponent n appeared<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 major advances in the history of the Great Theorem were observed. But soon an interesting event took place in mathematical life. In 1955, the 28-year-old Japanese mathematician Yutaka Taniyama advanced a statement from a completely different area of mathematics, called the Taniyama Hypothesis (aka the Taniyama-Shimura-Weil Hypothesis), which, unlike Fermat's belated Theorem, was ahead of its time.
Taniyama's conjecture states: "to every elliptic curve there corresponds a certain modular form." This statement for mathematicians of that time sounded about as absurd as the statement sounds for us: "a certain metal corresponds to each tree." It is easy to guess how a normal person can relate to such a statement - he simply will not take it seriously, which happened: mathematicians 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, fragments can be swapped, rotated in infinitely many ways - and their appearance does not change. As you can see, elliptic curves and modular forms have little in common. Taniyama's hypothesis states that the descriptive equations of these two absolutely different mathematical objects corresponding to each other can be expanded into the same mathematical series.
Taniyama's hypothesis was too paradoxical: it combined completely different concepts - rather simple flat curves and unimaginable four-dimensional shapes. This never occurred to anyone. When, at an international mathematical symposium in Tokyo in September 1955, Taniyama demonstrated several correspondences between elliptic curves and modular forms, everyone saw this as nothing more than a funny coincidence. To Taniyama's modest question: is it possible to find the corresponding modular function for each elliptic curve, the venerable Frenchman Andre Weil, who at that time was one of the world's best specialists in number theory, gave a quite diplomatic answer, what, 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 grown up to it yet - 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 - an 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. Recently, I often heard from others that I was tired mentally and physically. Actually, even now I don’t understand why I’m doing this ...” and so on on three sheets. It’s a pity, of course, that this was 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 much use as a telescope in a theater”) . The hypothesis has been abandoned. Nobody knew how to prove it.
For ten years, Taniyama's hypothesis was hardly mentioned. But in the early 70s, it became popular - it was regularly checked by everyone who could understand it - and it was always confirmed (as, in fact, Fermat's theorem), but, as before, no one could prove it.
The amazing connection between the two hypotheses
Another 15 years have passed. In 1984, there was one key event in the life of mathematics that combined the extravagant Japanese conjecture with Fermat's Last Theorem. The German Gerhard Frey put forward a curious statement, similar to a theorem: "If Taniyama's conjecture is proved, then, consequently, Fermat's Last Theorem will be proved." In other words, Fermat's theorem is a consequence of Taniyama's conjecture. (Frey, using ingenious 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), a professor at 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 exactly a consequence of Taniyama's conjecture, all that is needed is to prove the latter in order to break the laurels of the conqueror of the legendary Fermat's theorem. But the hypothesis turned out to be difficult. In addition, over the centuries, mathematicians became allergic to Fermat's theorem, and many of them decided that it would also be almost impossible to cope with Taniyama's conjecture.
The death of Fermat's hypothesis. The birth of a 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, therefore, looks like a genius. Entering into History, of course, is tempting and very desirable, but Wiles, like a real scientist, did not flatter himself, realizing that thousands of Fermists 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 involved others in 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 but his wife knew he was working on proving the Great Theorem.
And so, after long checks and painful reflections, Wiles finally plucked up courage, and perhaps, as he himself thought, arrogance, and on June 23, 1993, at a mathematical conference on number theory in Cambridge, he announced his great achievement.
It was, of course, a sensation. No one expected such agility from a little-known mathematician. Then the press came along. Everyone was tormented by a burning interest. Slender formulas, like the strokes of a beautiful picture, appeared before the curious eyes of the audience. Real mathematicians, after all, they are like that - they look at all sorts of equations and see in them not numbers, constants and variables, but they hear music, like Mozart looking at a musical staff. Just like when we read a book, we look at the letters, but we don’t seem to 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 simply stared at him like first-graders at an integral and did not understand anything). Everyone decided that a large-scale event had happened: Taniyama's hypothesis was proved, and consequently Fermat's Last Theorem. But about two months later, a few days before the manuscript of Wiles's proof was to go into circulation, it was found to be inconsistent (Katz, a colleague of Wiles, noted that one piece of reasoning relied on "Euler's system", but what built by Wiles, was not such a system), although, in general, Wiles's techniques were considered interesting, elegant and innovative.
Wiles analyzed the situation and decided that he had lost. One can imagine how he felt with all his being what it means "from the great to the ridiculous one step." "I wanted to enter History, but instead I joined a team of clowns and comedians - arrogant farmists" - approximately such thoughts exhausted him during that painful period of his life. For him, a serious 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 together with his colleague Taylor from Oxford, the latter suddenly had the idea that the "Euler system" could be changed to the Iwasawa theory (section of number theory). Then they tried to use the Iwasawa theory, doing without 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, in one of the leading mathematical journals - "Annals of Mathematics" - a complete proof of Taniyama's conjecture (hence, Fermat's Great (Large) Theorem) was published, which occupied the entire issue - over one 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 20th century, the whole world recognized that in the 360th year of its life, Fermat's Last Theorem, which in fact had been a hypothesis all this time, had become a proven theorem. Andrew Wiles proved Fermat's Great (Great) Theorem and entered History.
Think you've proven a theorem...
The happiness of the discoverer always goes to someone alone - it is he who, with the last blow of the hammer, cracks the hard nut of knowledge. But one cannot ignore the many previous blows that have formed a crack in the Great Theorem for centuries: Euler and Gauss (the kings of mathematics of their time), Evariste Galois (who managed to establish the theory of groups and fields in his short 21-year life, whose works were recognized as brilliant only after his death), Henri Poincaré (the founder of not only bizarre modular forms, but also conventionalism - a philosophical trend), David Gilbert (one of the strongest mathematicians of the twentieth century), Yutaku Taniyama, Goro Shimura, 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 not so widely known about it, because it does not invade the zone of our momentary interests, such as a TV set or an electric light bulb, it was a flash of a supernova, which, like all immutable truths, will always shine on humanity.
You can say: "Just think, you proved some kind of theorem, who needs it?". A fair question. David Gilbert's answer will fit exactly here. When, to the question: "what is the most important task for science now?", He answered: "to catch a fly on the far side of the moon", he was reasonably asked: "but who needs it?", he replied like this:" Nobody needs it. But think about how many important and difficult problems need to be solved in order to accomplish this. "Think about how many problems humanity has been able to solve in 360 years before proving Fermat's theorem. In search of its proof, almost half of modern mathematics was discovered. We must also take into account that mathematics is the avant-garde of science (and, by the way, the only one of the sciences that is built without a single mistake), and any scientific achievements and inventions begin here. ".
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And now let's go back to the beginning of our story, remember Pierre Fermat's entry in the margins of Diophantus's textbook and once again ask ourselves the question: did Fermat really prove his theorem? Of course, we cannot know this for sure, and as in any case, different versions arise here:
Version 1: Fermat proved his theorem. (To the question: "Did Fermat have exactly the same proof of his theorem?", Andrew Wiles remarked: "Fermat could not have so proof. This is the proof of the 20th century. "We understand that in the 17th century mathematics, of course, was not the same as at the end of the 20th century - in that era, d, Artagnan, the queen of sciences, did not yet possess those discoveries (modular forms, Taniyama's theorems , Frey, etc.), which only made it possible 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, is practically impossible according to most mathematicians);
Version 2: It seemed to Pierre de Fermat that he had proved his theorem, but there were errors in his proof. (That is, Fermat himself was also the first Fermatist);
Version 3: Fermat did not prove his theorem, but simply lied in the margins.
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 make mistakes or sometimes do not mind lying(basically, this conclusion will be useful for those who are inclined to completely trust their idols and other rulers of thoughts). Therefore, when reading the works of authoritative sons of mankind or listening to their pathetic speeches, you have every right to doubt their statements. (Please note that to doubt is not to reject).
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