Ivliev Yu.A.
The article is devoted to the description of a fundamental mathematical error made in the process of proving Fermat's Last Theorem at the end of the 20th century. The detected error not only distorts the true meaning of the theorem, but also hinders the development of a new axiomatic approach to the study of powers of numbers and the natural series of numbers.
In 1995, an article was published that was similar in size to a book and reported on the proof of the famous Fermat's Great (Last) Theorem (WTF) (for the history of the theorem and attempts to prove it, see, for example,). After this event, there were many scientific articles and popular science books promoting this proof, however, in none of these works was a fundamental mathematical error revealed in it, which crept in not even through the fault of the author, but due to some strange optimism that seized the minds of mathematicians who dealt with this problem and related questions with her. The psychological aspects of this phenomenon have been investigated in. It also gives a detailed analysis of the oversight that occurred, which is not of a particular nature, but is the result of an incorrect understanding of the properties of the powers of integers. As shown in , Fermat's problem is rooted in a new axiomatic approach to the study of these properties, which is still in modern science was not applied. But an erroneous proof stood in his way, giving number theorists false guidelines and leading researchers of Fermat's problem away from its direct and adequate solution. This work is devoted to removing this obstacle.
1. Anatomy of a mistake made during the proof of the WTF
In the process of very long and tedious reasoning, Fermat's original statement was reformulated in terms of a correspondence between a Diophantine equation of the p-th degree and elliptic curves of the 3rd order (see Theorems 0.4 and 0.5 in ). Such a comparison forced the authors of the de facto collective proof to announce that their method and reasoning lead to the final solution of Fermat's problem (recall that the WTF did not have recognized proofs for the case of arbitrary integer powers of integers until the 90s of the last century). The purpose of this consideration is to establish the mathematical incorrectness of the above comparison and, as a result of the analysis, to find a fundamental error in the proof presented in .
a) Where and what is wrong?
So, let's go through the text, where on p.448 it is said that after the "witty idea" of G. Frey (G. Frey), the possibility of proving the WTF has opened up. In 1984, G. Frey suggested and
K.Ribet later proved that the putative elliptic curve representing the hypothetical integer solution of Fermat's equation,
y 2 = x(x + u p)(x - v p) (1)
cannot be modular. However, A.Wiles and R.Taylor proved that any semistable elliptic curve defined over the field of rational numbers is modular. This led to the conclusion about the impossibility of integer solutions of Fermat's equation and, consequently, the validity of Fermat's statement, which in the notation of A. Wiles was written as Theorem 0.5: let there be an equality
u p+ v p+ w p = 0 (2)
where u, v, w- rational numbers, integer exponent p ≥ 3; then (2) is satisfied only if uvw = 0 .
Now, apparently, we should go back and critically consider why the curve (1) was a priori perceived as elliptic and what is its real relationship with Fermat's equation. Anticipating this question, A. Wiles refers to the work of Y. Hellegouarch, in which he found a way to associate Fermat's equation (presumably solved in integers) with a hypothetical 3rd order curve. Unlike G. Frey, I. Allegouches did not connect his curve with modular forms, but his method of obtaining equation (1) was used to further advance the proof of A. Wiles.
Let's take a closer look at work. The author conducts his reasoning in terms of projective geometry. Simplifying some of its notation and bringing them into line with , we find that the Abelian curve
Y 2 = X(X - β p)(X + γ p) (3)
the Diophantine equation is compared
x p+ y p+ z p = 0 (4)
where x, y, z are unknown integers, p is an integer exponent from (2), and the solutions of the Diophantine equation (4) α p , β p , γ p are used to write the Abelian curve (3).
Now, to make sure that this is a 3rd order elliptic curve, it is necessary to consider the variables X and Y in (3) on the Euclidean plane. To do this, we use the well-known rule of arithmetic of elliptic curves: if there are two rational points on a cubic algebraic curve and the line passing through these points intersects this curve at one more point, then the latter is also a rational point. Hypothetical equation (4) formally represents the law of addition of points on a straight line. If we make a change of variables x p = A, y p=B, z p = C and direct the straight line thus obtained along the X axis in (3), then it will intersect the 3rd degree curve at three points: (X = 0, Y = 0), (X = β p , Y = 0), (X = - γ p , Y = 0), which is reflected in the notation of the Abelian curve (3) and in a similar notation (1). However, is curve (3) or (1) really elliptical? Obviously not, because the segments of the Euclidean line, when adding points on it, are taken on a non-linear scale.
Returning to the linear coordinate systems of the Euclidean space, instead of (1) and (3) we obtain formulas that are very different from the formulas for elliptic curves. For example, (1) could be of the following form:
η 2p = ξ p (ξ p + u p)(ξ p - v p) (5)
where ξ p = x, η p = y, and the appeal to (1) in this case for the derivation of the WTF seems to be illegal. Despite the fact that (1) satisfies some criteria of the class of elliptic curves, it does not satisfy the most important criterion of being a 3rd degree equation in a linear coordinate system.
b) Error classification
So, once again we return to the beginning of the consideration and follow how the conclusion about the truth of the WTF is made. First, it is assumed that there is a solution of Fermat's equation in positive integers. Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (a plane curve of the 3rd degree) under the assumption that the elliptic curves thus obtained exist (the second unverified assumption). Thirdly, since it is proved by other methods that the constructed concrete curve is non-modular, it means that it does not exist. The conclusion follows from this: there is no integer solution of the Fermat equation and, therefore, the WTF is true.
There is one weak link in these arguments, which, after a detailed check, turns out to be a mistake. This mistake is made at the second stage of the proof process, when it is assumed that the hypothetical solution of Fermat's equation is also the solution of a third-degree algebraic equation describing an elliptic curve of a known form. In itself, such an assumption would be justified if the indicated curve were indeed elliptic. However, as can be seen from item 1a), this curve is presented in non-linear coordinates, which makes it “illusory”, i.e. not really existing in a linear topological space.
Now we need to clearly classify the found error. It lies in the fact that what needs to be proved is given as an argument of the proof. In classical logic, this error is known as the "vicious circle". In this case, the integer solution of the Fermat equation is compared (apparently, presumably uniquely) with a fictitious, non-existent elliptic curve, and then all the pathos of further reasoning goes to prove that a specific elliptic curve of this form, obtained from hypothetical solutions of the Fermat equation, does not exist.
How did it happen that such an elementary mistake was missed in a serious mathematical work? Probably, this happened due to the fact that “illusory” concepts were not previously studied in mathematics. geometric figures the specified type. Indeed, who could be interested, for example, in a fictitious circle obtained from Fermat's equation by changing the variables x n/2 = A, y n/2 = B, z n/2 = C? After all, its equation C 2 = A 2 + B 2 has no integer solutions for integer x, y, z and n ≥ 3 . In non-linear coordinate axes X and Y, such a circle would be described by an equation that looks very similar to the standard form:
Y 2 \u003d - (X - A) (X + B),
where A and B are no longer variables, but concrete numbers determined by the above substitution. But if the numbers A and B are given their original form, which consists in their power character, then the heterogeneity of the notation in the factors on the right side of the equation immediately catches the eye. This sign helps to distinguish illusion from reality and to move from non-linear to linear coordinates. On the other hand, if we consider numbers as operators when comparing them with variables, as for example in (1), then both must be homogeneous quantities, i.e. must have the same degree.
Such an understanding of the powers of numbers as operators also makes it possible to see that the comparison of Fermat's equation with an illusory elliptic curve is not unambiguous. Take, for example, one of the factors on the right side of (5) and expand it into p linear factors by introducing a complex number r such that r p = 1 (see for example ):
ξ p + u p = (ξ + u)(ξ + r u)(ξ + r 2 u)...(ξ + r p-1 u) (6)
Then the form (5) can be represented as a decomposition into prime factors of complex numbers according to the type of algebraic identity (6), however, the uniqueness of such a decomposition in the general case is questionable, which was once shown by Kummer.
2. Conclusions
It follows from the previous analysis that the so-called arithmetic of elliptic curves is not capable of shedding light on where to look for the proof of the WTF. After the work, Fermat's statement, by the way, taken as the epigraph to this article, began to be perceived as a historical joke or practical joke. However, in reality it turns out that it was not Fermat who was joking, but experts who gathered at a mathematical symposium in Oberwolfach in Germany in 1984, at which G. Frey voiced his witty idea. The consequences of such a careless statement brought mathematics as a whole to the verge of losing its public confidence, which is described in detail in and which necessarily raises the question of the responsibility of scientific institutions to society before science. The mapping of the Fermat equation to the Frey curve (1) is the "lock" of Wiles's entire proof with respect to Fermat's theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof either.
Lately there have been various Internet reports that some prominent mathematicians have finally figured out Wiles' proof of Fermat's theorem, giving him an excuse in the form of a "minimal" recalculation of integer points in Euclidean space. However, no innovations can cancel the classical results already obtained by mankind in mathematics, in particular, the fact that although any ordinal number coincides with its quantitative counterpart, it cannot be a replacement for it in operations of comparing numbers with each other, and hence with inevitably follows the conclusion that the Frey curve (1) is not elliptic initially, i.e. is not by definition.
BIBLIOGRAPHY:
- Ivliev Yu.A. Reconstruction of the native proof of Fermat's Last Theorem - United Scientific Journal (section "Mathematics"). April 2006 No. 7 (167) p.3-9, see also Pratsi of the Luhansk branch of the International Academy of Informatization. Ministry of Education and Science of Ukraine. Shidnoukrainian National University named after. V. Dahl. 2006 No. 2 (13) pp.19-25.
- Ivliev Yu.A. The greatest scientific scam of the 20th century: the "proof" of Fermat's Last Theorem - Natural and technical sciences (section "History and methodology of mathematics"). August 2007 No. 4 (30) pp. 34-48.
- Edwards G. (Edwards H.M.) Fermat's last theorem. Genetic introduction to algebraic number theory. Per. from English. ed. B.F. Skubenko. M.: Mir 1980, 484 p.
- Hellegouarch Y. Points d'ordre 2p h sur les courbes elliptiques - Acta Arithmetica. 1975 XXVI p.253-263.
- Wiles A. Modular elliptic curves and Fermat´s Last Theorem - Annals of Mathematics. May 1995 v.141 Second series No. 3 p.443-551.
Bibliographic link
Ivliev Yu.A. WILES' ERROROUS PROOF OF THE GREAT THEOREM OF FARM // Basic Research. - 2008. - No. 3. - P. 13-16;URL: http://fundamental-research.ru/ru/article/view?id=2763 (date of access: 09/25/2019). We bring to your attention the journals published by the publishing house "Academy of Natural History"
So, The Great Theorem Fermat (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 ordinary 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 the 5th grade 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 all sides are equal." 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 Pythagorean triplets infinitely many, 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, and so on. What if we take a similar equation x³+y³=z³ ? Maybe there are such numbers too?
And so on (Fig. 1).
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 explored general equation x n+yn=zn . 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 accommodate 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, great minds such as Leonhard Euler worked on finding the 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-1980s, it became clear that academia is on the way to the final solution of Fermat's Last Theorem, but it was not until 1993 that mathematicians saw and believed 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 conjecture. 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 conjecture, 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 causes too much great interest… Too many spectators obviously 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 ...
FERMAT GREAT THEOREM - the statement of Pierre Fermat (French lawyer and part-time mathematician) that the Diophantine equation X n + Y n \u003d Z n, with an exponent n>2, where n = an integer, has no solutions in positive integers . Author's text: "It is impossible to decompose a cube into two cubes, or a bi-square into two bi-squares, or in general a power greater than two into two powers with the same exponent."
"Fermat and his theorem", Amadeo Modigliani, 1920
Pierre came up with this theorem on March 29, 1636. And after some 29 years, he died. But that's where it all started. After all, a wealthy German mathematician by the name of Wolfskel bequeathed one hundred thousand marks to the one who presents the complete proof of Fermat's theorem! But the excitement around the theorem was connected not only with this, but also with professional mathematical excitement. Fermat himself hinted to the mathematical community that he knew the proof - shortly before his death, in 1665, he left the following entry in the margins of the book Diophantus of Alexandria "Arithmetic": "I have a very amazing proof, but it is too large to be placed on fields."
It was this hint (plus, of course, a cash prize) that made mathematicians unsuccessfully spend their best years searching for proof (according to American scientists, professional mathematicians alone spent 543 years on this in total).
At some point (in 1901), work on Fermat's theorem acquired the dubious fame of "work akin to the search for a perpetual motion machine" (there was even a derogatory term - "fermatists"). And suddenly, on June 23, 1993, at a mathematical conference on number theory in Cambridge, English professor of mathematics from Princeton University (New Jersey, USA) Andrew Wiles announced that he had finally proved Fermat!
The proof, however, was not only complicated, but also obviously erroneous, as Wiles was pointed out by his colleagues. But Professor Wiles dreamed of proving the theorem all his life, so it is not surprising that in May 1994 he presented a new, improved version of the proof to the scientific community. There was no harmony, beauty in it, and it was still very complicated - the fact that mathematicians have been analyzing this proof for a whole year (!) To understand whether it is not erroneous, speaks for itself!
But in the end, Wiles' proof was found to be correct. But mathematicians did not forgive Pierre Fermat for his very hint in Arithmetic, and, in fact, they began to consider him a liar. In fact, the first person to question Fermat's moral integrity was Andrew Wiles himself, who remarked that "Fermat could not have had such proof. This is twentieth-century proof." Then, among other scientists, the opinion became stronger that Fermat "could not prove his theorem in another way, and Fermat could not prove it in the way that Wiles went, for objective reasons."
In fact, Fermat, of course, could prove it, and a little later this proof will be recreated by the analysts of the New Analytical Encyclopedia. But - what are these "objective reasons"?
In fact, there is only one such reason: in those years when Fermat lived, Taniyama’s conjecture could not appear, on which Andrew Wiles built his proof, because the modular functions that Taniyama’s conjecture operates on were discovered only in late XIX century.
How did Wiles himself prove the theorem? The question is not idle - this is important for understanding how Fermat himself could prove his theorem. Wiles built his proof on the proof of Taniyama's conjecture put forward in 1955 by the 28-year-old Japanese mathematician Yutaka Taniyama.
The conjecture sounds like this: "every elliptic curve corresponds to a certain modular form." Elliptic curves, known for a long time, have a two-dimensional form (located on a plane), while modular functions have a four-dimensional form. That is, Taniyama's hypothesis combined completely different concepts - simple flat curves and unimaginable four-dimensional forms. The very fact of connecting different-dimensional figures in the hypothesis seemed absurd to scientists, which is why in 1955 it was not given any importance.
However, in the fall of 1984, the "Taniyama hypothesis" was suddenly remembered again, and not only remembered, but its possible proof was connected with the proof of Fermat's theorem! This was done by Saarbrücken mathematician Gerhard Frey, who told the scientific community that "if anyone could prove Taniyama's conjecture, then Fermat's Last Theorem would be proved."
What did Frey do? He converted Fermat's equation to a cubic one, then drew attention to the fact that an elliptic curve obtained by converting Fermat's equation to a cubic one cannot be modular. However, Taniyama's conjecture stated that any elliptic curve could be modular! Accordingly, an elliptic curve constructed from Fermat's equation cannot exist, which means there cannot be entire solutions and Fermat's theorem, which means it is true. Well, in 1993, Andrew Wiles simply proved Taniyama's conjecture, and hence Fermat's theorem.
However, Fermat's theorem can be proved much more simply, on the basis of the same multidimensionality that both Taniyama and Frey operated on.
To begin with, let's pay attention to the condition stipulated by Pierre Fermat himself - n>2. Why was this condition necessary? Yes, only for the fact that for n=2 the ordinary Pythagorean theorem X 2 +Y 2 =Z 2 becomes a special case of Fermat's theorem, which has an infinite number of integer solutions - 3,4,5; 5,12,13; 7.24.25; 8,15,17; 12,16,20; 51,140,149 and so on. Thus, the Pythagorean theorem is an exception to Fermat's theorem.
But why exactly in the case of n=2 does such an exception occur? Everything falls into place if you see the relationship between the degree (n=2) and the dimension of the figure itself. The Pythagorean triangle is a two-dimensional figure. Not surprisingly, Z (that is, the hypotenuse) can be expressed in terms of legs (X and Y), which can be integers. The size of the angle (90) makes it possible to consider the hypotenuse as a vector, and the legs are vectors located on the axes and coming from the origin. Accordingly, it is possible to express a two-dimensional vector that does not lie on any of the axes in terms of the vectors that lie on them.
Now, if we go to the third dimension, and hence to n=3, in order to express a three-dimensional vector, there will not be enough information about two vectors, and therefore it will be possible to express Z in Fermat's equation through at least three terms (three vectors lying, respectively, on the three axes of the coordinate system).
If n=4, then there should be 4 terms, if n=5, then there should be 5 terms, and so on. In this case, there will be more than enough whole solutions. For example, 3 3 +4 3 +5 3 =6 3 and so on (you can choose other examples for n=3, n=4 and so on).
What follows from all this? It follows from this that Fermat's theorem does indeed have no entire solutions for n>2 - but only because the equation itself is incorrect! With the same success, one could try to express the volume of a parallelepiped in terms of the lengths of its two edges - of course, this is impossible (whole solutions will never be found), but only because to find the volume of a parallelepiped, you need to know the lengths of all three of its edges.
When the famous mathematician David Gilbert was asked what is the most important task for science now, he answered "to catch a fly on the far side of the moon." To the reasonable question "Who needs it?" he replied: "Nobody needs this. But think about how many important the most difficult tasks You have to decide to make it happen."
In other words, Fermat (a lawyer in the first place!) played a witty legal joke on the entire mathematical world, based on an incorrect formulation of the problem. He, in fact, suggested that mathematicians find an answer why a fly cannot live on the other side of the Moon, and in the margins of Arithmetic he wanted to write only that there is simply no air on the Moon, i.e. there can be no integer solutions of his theorem for n>2 just because each value of n must correspond to a certain number of terms on the left side of his equation.
But was it just a joke? Not at all. Fermat's genius lies precisely in the fact that he was actually the first to see the relationship between degree and dimension. mathematical figure- that is, which is absolutely equivalent to the number of terms on the left side of the equation. The meaning of his famous theorem was precisely to not only push the mathematical world on the idea of this relationship, but also to initiate a proof of the existence of this relationship - intuitively understandable, but not yet mathematically substantiated.
Fermat, like no one else, understood that establishing a relationship between seemingly different objects is extremely fruitful not only in mathematics, but also in any science. Such a relationship points to some deep principle underlying both objects and allowing a deeper understanding of them.
For example, initially physicists considered electricity and magnetism as completely unrelated phenomena, and in the 19th century, theorists and experimenters realized that electricity and magnetism were closely related. The result was a deeper understanding of both electricity and magnetism. Electric currents generate magnetic fields, and magnets can induce electricity in conductors that are close to the magnets. This led to the invention of dynamos and electric motors. Eventually it was discovered that light is the result of coordinated harmonic oscillations of magnetic and electric fields.
The mathematics of Fermat's time consisted of islands of knowledge in a sea of ignorance. Geometers studied shapes on one island, and mathematicians studied probability and chance on the other island. The language of geometry was very different from the language of probability theory, and algebraic terminology was alien to those who spoke only about statistics. Unfortunately, mathematics of our time consists of approximately the same islands.
Farm was the first to realize that all these islands are interconnected. And his famous theorem - Fermat's GREAT THEOREM - is an excellent confirmation of this.
Envious people claim that the French mathematician Pierre Fermat entered his name in history with just one phrase. In the margin of the manuscript with the formulation of the famous theorem in 1637, he made a note: "I found an amazing solution, but there is not enough space to put it." Then an amazing mathematical race began, in which, along with outstanding scientists, an army of amateurs joined.
What is the insidiousness of Fermat's problem? At first glance, it is clear even to a schoolboy.
It is based on the well-known Pythagorean theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: x 2 + y 2 \u003d z 2. Fermat argued that an equation with any powers greater than two has no solution in integers.
It would seem simple. Reach out your hand and here is the answer. Not surprisingly, the academies different countries, scientific institutes, even newspaper editorial offices were inundated with tens of thousands of evidence. Their number is unprecedented, second only to projects of "perpetual motion machines". But if serious science has not considered these crazy ideas for a long time, then the works of "fermists" are honestly and interestedly studying. And, alas, finds errors. They say that in three superfluous century a whole mathematical cemetery of solutions of the theorem was formed.
No wonder they say: the elbow is close, but you won’t bite. Years, decades, centuries passed, and Fermat's problem seemed more and more surprising and tempting. It seems to be unpretentious, it turned out to be too tough for the progress that is rapidly building up muscles. Man has already split the atom, got to the gene, set foot on the moon, but Fermat did not give in, continuing to beckon his descendants with false hopes.
However, attempts to overcome the scientific pinnacle were not in vain. The first step was taken by the great Euler, proving the theorem for the fourth degree, then for the third. At the end of the 19th century, the German Ernst Kummer brought the number of degrees to one hundred. Finally, armed with computers, scientists increased this figure to 100,000. But Fermat spoke of any degrees. That was the whole point.
Of course, scientists were tormented by the task not because of sports interest. The famous mathematician David Hilbert said that a theorem is an example of how a seemingly insignificant problem can have a huge impact on science. Working on it, scientists opened up completely new mathematical horizons, for example, the foundations of number theory, algebra, and function theory were laid.
And yet the Great Theorem was subdued in 1995. Her solution was presented by an American from Princeton University, Andrew Wiles, and it is officially recognized by the scientific community. He gave more than seven years of his life to find proof. According to scientists, this outstanding work brought together the works of many mathematicians, restoring the lost links between its different sections.
So, the summit has been taken, and science has received an answer, - the scientific secretary of the Department of Mathematics told the RG correspondent Russian Academy sciences, doctor technical sciences Yuri Vishnyakov. - The theorem has been proved, albeit not in the simplest way, as Fermat himself insisted on. And now those who wish can print their own versions.
However, the "fermist" family is not going to accept Wiles' proof at all. No, they do not refute the American's decision, because it is very complex, and therefore understandable only to a narrow circle of specialists. But not a week goes by without a new revelation of another enthusiast appearing on the Internet, "finally putting an end to a long-term epic."
By the way, just yesterday, one of the oldest "fermists" in our country, Vsevolod Yarosh, called the editorial office of "RG": "Do you know that I proved Fermat's theorem even before Wiles. Moreover, later I found a mistake in him, about which I wrote to our outstanding Mathematician Academician Arnold with a request to publish this in a scientific journal. Now I am waiting for an answer. I am also corresponding with the French Academy of Sciences on this matter. "
And just now, as reported in a number of media outlets, with "light grace he revealed the great secret of mathematics," another enthusiast is the former general designer of the Polet software from Omsk, Doctor of Technical Sciences Alexander Ilyin. The solution turned out to be so simple and short that it fit on a small section of the newspaper area of one of the central publications.
The editors of "RG" turned to the country's leading Institute of Mathematics. Steklov RAS with a request to evaluate this solution. Scientists were categorical: you can not comment on a newspaper publication. But after much persuasion and taking into account the increased interest in the famous problem, they agreed. According to them, several fundamental errors were made in the published proof. By the way, even a student of the Faculty of Mathematics could have noticed them.
And yet the editors wanted to get first-hand information. Moreover, yesterday at the Academy of Aviation and Aeronautics, Ilyin was supposed to present his proof. However, it turned out that few people even among specialists know about such an academy. And when, nevertheless, with great difficulty, it was possible to find the telephone number of the scientific secretary of this organization, then, as it turned out, he did not even suspect that such a historical event was to take place in their place. In a word, the correspondent of "RG" did not succeed in becoming a witness to the world sensation.
It is unlikely that at least one year in the life of our editorial office passed without it receiving a good dozen proofs of Fermat's theorem. Now, after the “victory” over it, the flow has subsided, but has not dried up.
Of course, not to dry it completely, we publish this article. And not in my own defense - that, they say, that's why we kept silent, we ourselves have not matured yet to discuss such complex problems.
But if the article really seems complicated, look at the end of it right away. You will have to feel that the passions have calmed down temporarily, the science is not over, and soon new proofs of new theorems will be sent to the editors.
It seems that the 20th century was not in vain. First, people created a second Sun for a moment by detonating a hydrogen bomb. Then they walked on the moon and finally proved the notorious Fermat's theorem. Of these three miracles, the first two are on everyone's lips, for they have had enormous social consequences. On the contrary, the third miracle looks like another scientific toy - on a par with the theory of relativity, quantum mechanics and Gödel's theorem on the incompleteness of arithmetic. However, relativity and quanta led physicists to the hydrogen bomb, and the research of mathematicians filled our world with computers. Will this string of miracles continue into the 21st century? Is it possible to trace the connection between the next scientific toys and revolutions in our everyday life? Does this connection allow us to make successful predictions? Let's try to understand this using the example of Fermat's theorem.
Let's note for a start that she was born much later than her natural term. After all, the first special case of Fermat's theorem is the Pythagorean equation X 2 + Y 2 = Z 2 , relating the lengths of the sides of a right triangle. Having proved this formula twenty-five centuries ago, Pythagoras immediately asked himself the question: are there many triangles in nature in which both legs and hypotenuse have an integer length? It seems that the Egyptians knew only one such triangle - with sides (3, 4, 5). But it is not difficult to find other options: for example (5, 12, 13) , (7, 24, 25) or (8, 15, 17) . In all these cases, the length of the hypotenuse has the form (A 2 + B 2), where A and B are coprime numbers of different parity. In this case, the lengths of the legs are equal to (A 2 - B 2) and 2AB.
Noticing these relationships, Pythagoras easily proved that any triple of numbers (X \u003d A 2 - B 2, Y \u003d 2AB, Z \u003d A 2 + B 2) is a solution to the equation X 2 + Y 2 \u003d Z 2 and sets a rectangle with mutually simple side lengths. It is also seen that the number of different triples of this sort is infinite. But do all solutions of the Pythagorean equation have this form? Pythagoras was unable to prove or disprove such a hypothesis and left this problem to posterity without drawing attention to it. Who wants to highlight their failures? Looks like after this the problem of integers right triangles lay in oblivion for seven centuries - until a new mathematical genius named Diophantus appeared in Alexandria.
We know little about him, but it is clear that he was nothing like Pythagoras. He felt like a king in geometry and even beyond - whether in music, astronomy or politics. The first arithmetic connection between the lengths of the sides of a harmonious harp, the first model of the Universe from concentric spheres carrying planets and stars, with the Earth in the center, and finally, the first republic of scientists in the Italian city of Crotone - these are the personal achievements of Pythagoras. What could Diophantus oppose to such successes - a modest researcher of the great Museum, which has long ceased to be the pride of the city crowd?
Only one thing: a better understanding ancient world numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet master the positional notation of large numbers, but he knew what negative numbers were and probably spent many hours thinking about why the product of two negative numbers is positive. The world of integers was first revealed to Diophantus as a special universe, different from the world of stars, segments or polyhedra. The main occupation of scientists in this world is solving equations, a true master finds all possible solutions and proves that there are no other solutions. This is what Diophantus did quadratic equation Pythagoras, and then he thought: does at least one solution have a similar cubic equation X 3 + Y 3 = Z 3?
Diophantus failed to find such a solution; his attempt to prove that there are no solutions was also unsuccessful. Therefore, drawing up the results of his work in the book "Arithmetic" (it was the world's first textbook on number theory), Diophantus analyzed the Pythagorean equation in detail, but did not hint at a word about possible generalizations of this equation. But he could: after all, it was Diophantus who first proposed the notation for the powers of integers! But alas: the concept of “task book” was alien to Hellenic science and pedagogy, and publishing lists of unsolved problems was considered an indecent occupation (only Socrates acted differently). If you can't solve the problem - shut up! Diophantus fell silent, and this silence dragged on for fourteen centuries - until the onset of the New Age, when interest in the process of human thinking was revived.
Who didn’t fantasize about anything at the turn of the 16th-17th centuries! The indefatigable calculator Kepler tried to guess the connection between the distances from the Sun to the planets. Pythagoras failed. Kepler's success came after he learned how to integrate polynomials and other simple functions. On the contrary, the dreamer Descartes did not like long calculations, but it was he who first presented all points of the plane or space as sets of numbers. This audacious model reduces any geometric problem about figures to some algebraic problem about equations - and vice versa. For example, integer solutions of the Pythagorean equation correspond to integer points on the surface of a cone. The surface corresponding to the cubic equation X 3 + Y 3 = Z 3 looks more complicated, its geometric properties did not suggest anything to Pierre Fermat, and he had to pave new paths through the wilds of integers.
In 1636, a book by Diophantus, just translated into Latin from a Greek original, fell into the hands of a young lawyer from Toulouse, accidentally surviving in some Byzantine archive and brought to Italy by one of the Roman fugitives at the time of the Turkish ruin. Reading an elegant discussion of the Pythagorean equation, Fermat thought: is it possible to find such a solution, which consists of three square numbers? There are no small numbers of this kind: it is easy to verify this by enumeration. What about big decisions? Without a computer, Fermat could not carry out a numerical experiment. But he noticed that for each "large" solution of the equation X 4 + Y 4 = Z 4, one can construct a smaller solution. So the sum of the fourth powers of two integers is never equal to the same power of the third number! What about the sum of two cubes?
Inspired by the success for degree 4, Fermat tried to modify the "method of descent" for degree 3 - and succeeded. It turned out that it was impossible to compose two small cubes from those single cubes into which a large cube with an integer length of an edge fell apart. The triumphant Fermat made a brief note in the margins of Diophantus's book and sent a letter to Paris with a detailed report of his discovery. But he did not receive an answer - although usually mathematicians from the capital reacted quickly to the next success of their lone colleague-rival in Toulouse. What's the matter here?
Quite simply: by the middle of the 17th century, arithmetic had gone out of fashion. The great successes of the Italian algebraists of the 16th century (when polynomial equations of degrees 3 and 4 were solved) did not become the beginning of a general scientific revolution, because they did not allow solving new bright problems in adjacent fields of science. Now, if Kepler could guess the orbits of the planets using pure arithmetic ... But alas, this required mathematical analysis. This means that it must be developed - up to the complete triumph of mathematical methods in natural science! But analysis grows out of geometry, while arithmetic remains a field of play for idle lawyers and other lovers of the eternal science of numbers and figures.
So, Fermat's arithmetic successes turned out to be untimely and remained unappreciated. He was not upset by this: for the fame of a mathematician, the facts of differential calculus, analytic geometry and probability theory were revealed to him for the first time. All these discoveries of Fermat immediately entered the golden fund of the new European science, while number theory faded into the background for another hundred years - until it was revived by Euler.
This "king of mathematicians" of the 18th century was a champion in all applications of analysis, but he did not neglect arithmetic either, since new methods of analysis led to unexpected facts about numbers. Who would have thought that the infinite sum of inverse squares (1 + 1/4 + 1/9 + 1/16+…) is equal to π 2 /6? Who among the Hellenes could have foreseen that similar series would make it possible to prove the irrationality of the number π?
Such successes forced Euler to carefully reread the surviving manuscripts of Fermat (fortunately, the son of the great Frenchman managed to publish them). True, the proof of the “big theorem” for degree 3 has not been preserved, but Euler easily restored it just by pointing to the “descent method”, and immediately tried to transfer this method to the next prime degree - 5.
It wasn't there! In Euler's reasoning, complex numbers appeared that Fermat managed not to notice (such is the usual lot of discoverers). But the factorization of complex integers is a delicate matter. Even Euler did not fully understand it and put the "Fermat problem" aside, in a hurry to complete his main work - the textbook "Principles of Analysis", which was supposed to help every talented young man to stand on a par with Leibniz and Euler. The publication of the textbook was completed in St. Petersburg in 1770. But Euler did not return to Fermat's theorem, being sure that everything that his hands and mind touched would not be forgotten by the new scientific youth.
And so it happened: the Frenchman Adrien Legendre became Euler's successor in number theory. AT late XVIII century, he completed the proof of Fermat's theorem for degree 5 - and although he failed for large prime powers, he compiled another textbook on number theory. May its young readers surpass the author in the same way that the readers of the Mathematical Principles of Natural Philosophy surpassed the great Newton! Legendre was no match for Newton or Euler, but there were two geniuses among his readers: Carl Gauss and Evariste Galois.
Such a high concentration of geniuses was facilitated by the French Revolution, which proclaimed the state cult of Reason. After that, every talented scientist felt like Columbus or Alexander the Great, able to discover or conquer a new world. Many succeeded, which is why in the 19th century scientific and technological progress became the main driver of the evolution of mankind, and all reasonable rulers (starting with Napoleon) were aware of this.
Gauss was close in character to Columbus. But he (like Newton) did not know how to captivate the imagination of rulers or students with beautiful speeches, and therefore limited his ambitions to the sphere of scientific concepts. Here he could do whatever he wanted. For example, the ancient problem of the trisection of an angle for some reason cannot be solved with a compass and straightedge. With the help of complex numbers depicting points of the plane, Gauss translates this problem into the language of algebra - and obtains a general theory of the feasibility of certain geometric constructions. Thus, at the same time, a rigorous proof of the impossibility of constructing a regular 7- or 9-gon with a compass and a ruler appeared, and such a way of constructing a regular 17-gon, which the wisest geometers of Hellas did not dream of.
Of course, such success is not given in vain: one has to invent new concepts that reflect the essence of the matter. Newton introduced three such concepts: flux (derivative), fluent (integral) and power series. They were enough to create mathematical analysis and the first scientific model physical world including mechanics and astronomy. Gauss also introduced three new concepts: vector space, field, and ring. A new algebra grew out of them, subordinating Greek arithmetic and the theory of numerical functions created by Newton. It only remained to subordinate the logic created by Aristotle to algebra: then it would be possible, with the help of calculations, to prove the derivability or non-derivability of any scientific statements from a given set of axioms! For example, does Fermat's theorem derive from the axioms of arithmetic, or does Euclid's postulate of parallel lines derive from other axioms of planimetry?
Gauss did not have time to realize this daring dream - although he advanced far and guessed the possibility of the existence of exotic (non-commutative) algebras. Only the daring Russian Nikolai Lobachevsky managed to build the first non-Euclidean geometry, and the first non-commutative algebra (Group Theory) was managed by the Frenchman Evariste Galois. And only much later than the death of Gauss - in 1872 - the young German Felix Klein guessed that the variety of possible geometries can be brought into one-to-one correspondence with the variety of possible algebras. Simply put, every geometry is defined by its symmetry group - while general algebra studies all possible groups and their properties.
But such an understanding of geometry and algebra came much later, and the assault on Fermat's theorem resumed during Gauss's lifetime. He himself neglected Fermat's theorem out of the principle: it's not the king's business to solve individual problems that do not fit into the bright scientific theory! But the students of Gauss, armed with his new algebra and the classical analysis of Newton and Euler, reasoned differently. First, Peter Dirichlet proved Fermat's theorem for degree 7 using the ring of complex integers generated by the roots of this degree of unity. Then Ernst Kummer extended the Dirichlet method to ALL prime degrees (!) - it seemed to him in a rush, and he triumphed. But soon a sobering up came: the proof passes flawlessly only if every element of the ring is uniquely decomposed into prime factors! For ordinary integers, this fact was already known to Euclid, but only Gauss gave its rigorous proof. But what about the whole complex numbers?
According to the “principle of the greatest mischief”, there can and SHOULD occur an ambiguous factorization! As soon as Kummer learned how to calculate the degree of ambiguity by methods of mathematical analysis, he discovered this dirty trick in the ring for degree 23. Gauss did not have time to learn about this version of exotic commutative algebra, but Gauss's students grew a new beautiful Theory of Ideals in place of another dirty trick. True, this did not help much in solving Fermat's problem: only its natural complexity became clearer.
Throughout the 19th century, this ancient idol demanded more and more sacrifices from its admirers in the form of new complex theories. It is not surprising that by the beginning of the 20th century, believers became discouraged and rebelled, rejecting their former idol. The word "fermatist" has become a pejorative term among professional mathematicians. And although a considerable prize was assigned for the complete proof of Fermat's theorem, but its applicants were mostly self-confident ignoramuses. The strongest mathematicians of that time - Poincaré and Hilbert - defiantly eschewed this topic.
In 1900, Hilbert did not include Fermat's Theorem in the list of twenty-three major problems facing the mathematics of the twentieth century. True, he included in their series the general problem of the solvability of Diophantine equations. The hint was clear: follow the example of Gauss and Galois, create general theories new mathematical objects! Then one fine (but not predictable in advance) day, the old splinter will fall out by itself.
This is how the great romantic Henri Poincaré acted. Neglecting many "eternal" problems, all his life he studied the SYMMETRIES of certain objects of mathematics or physics: either functions of a complex variable, or trajectories of motion of celestial bodies, or algebraic curves or smooth manifolds (these are multidimensional generalizations of curved lines). The motive for his actions was simple: if two different objects have similar symmetries, it means that there is an internal relationship between them, which we are not yet able to comprehend! For example, each of the two-dimensional geometries (Euclid, Lobachevsky or Riemann) has its own symmetry group, which acts on the plane. But the points of the plane are complex numbers: in this way the action of any geometric group is transferred to the vast world of complex functions. It is possible and necessary to study the most symmetrical of these functions: AUTOMORPHOUS (which are subject to the Euclid group) and MODULAR (which are subject to the Lobachevsky group)!
There are also elliptic curves in the plane. They have nothing to do with the ellipse, but are given by equations of the form Y 2 = AX 3 + BX 2 + CX and therefore intersect with any straight line at three points. This fact allows us to introduce multiplication among the points of an elliptic curve - to turn it into a group. The algebraic structure of this group reflects the geometric properties of the curve; perhaps it is uniquely determined by its group? This question is worth studying, since for some curves the group of interest to us turns out to be modular, that is, it is related to the Lobachevsky geometry ...
This is how Poincaré reasoned, seducing the mathematical youth of Europe, but at the beginning of the 20th century these temptations did not lead to bright theorems or hypotheses. It turned out differently with Hilbert's call: to study the general solutions of Diophantine equations with integer coefficients! In 1922, the young American Lewis Mordell connected the set of solutions of such an equation (this is a vector space of a certain dimension) with the geometric genus of the complex curve that is given by this equation. Mordell came to the conclusion that if the degree of the equation is sufficiently large (more than two), then the dimension of the solution space is expressed in terms of the genus of the curve, and therefore this dimension is FINITE. On the contrary - to the power of 2, the Pythagorean equation has an INFINITE-DIMENSIONAL family of solutions!
Of course, Mordell saw the connection of his hypothesis with Fermat's theorem. If it becomes known that for every degree n > 2 the space of entire solutions of Fermat's equation is finite-dimensional, this will help to prove that there are no such solutions at all! But Mordell did not see any way to prove his hypothesis - and although he lived a long life, he did not wait for the transformation of this hypothesis into Faltings' theorem. This happened in 1983, in a completely different era, after the great successes of the algebraic topology of manifolds.
Poincaré created this science as if by accident: he wanted to know what three-dimensional manifolds are. After all, Riemann figured out the structure of all closed surfaces and got a very simple answer! If there is no such answer in a three-dimensional or multidimensional case, then you need to come up with a system of algebraic invariants of the manifold that determines its geometric structure. It is best if such invariants are elements of some groups - commutative or non-commutative.
Strange as it may seem, this audacious plan by Poincaré succeeded: it was carried out from 1950 to 1970 thanks to the efforts of a great many geometers and algebraists. Until 1950, there was a quiet accumulation of various methods for classifying manifolds, and after this date, a critical mass of people and ideas seemed to have accumulated and an explosion occurred, comparable to the invention of mathematical analysis in the 17th century. But the analytic revolution stretched for a century and a half, embracing creative biographies of four generations of mathematicians - from Newton and Leibniz to Fourier and Cauchy. On the contrary, the topological revolution of the 20th century was within twenty years, thanks to the large number of its participants. At the same time, a large generation of self-confident young mathematicians has emerged, suddenly left without work in their historical homeland.
In the seventies they rushed into the adjacent fields of mathematics and theoretical physics. Many have created their own scientific schools in dozens of universities in Europe and America. Many students of different ages and nationalities, with different abilities and inclinations, still circulate between these centers, and everyone wants to be famous for some discovery. It was in this pandemonium that Mordell's conjecture and Fermat's theorem were finally proven.
However, the first swallow, unaware of its fate, grew up in Japan in the hungry and unemployed post-war years. The name of the swallow was Yutaka Taniyama. In 1955, this hero turned 28 years old, and he decided (together with friends Goro Shimura and Takauji Tamagawa) to revive mathematical research in Japan. Where to begin? Of course, with overcoming isolation from foreign colleagues! So in 1955, three young Japanese hosted the first international conference on algebra and number theory in Tokyo. It was apparently easier to do this in Japan reeducated by the Americans than in Russia frozen by Stalin ...
Among the guests of honor were two heroes from France: Andre Weil and Jean-Pierre Serre. Here the Japanese were very lucky: Weil was the recognized head of the French algebraists and a member of the Bourbaki group, and the young Serre played a similar role among topologists. In heated discussions with them, the heads of the Japanese youth cracked, their brains melted, but in the end, such ideas and plans crystallized that could hardly have been born in a different environment.
One day, Taniyama approached Weil with a question about elliptic curves and modular functions. At first, the Frenchman did not understand anything: Taniyama was not a master of speaking English. Then the essence of the matter became clear, but Taniyama did not manage to give his hopes an exact formulation. All Weil could reply to the young Japanese was that if he were very lucky in terms of inspiration, then something sensible would grow out of his vague hypotheses. But while the hope for it is weak!
Obviously, Weil did not notice the heavenly fire in Taniyama's gaze. And there was fire: it seems that for a moment the indomitable thought of the late Poincaré moved into the Japanese! Taniyama came to believe that every elliptic curve is generated by modular functions - more precisely, it is "uniformized by a modular form". Alas, this exact wording was born much later - in Taniyama's conversations with his friend Shimura. And then Taniyama committed suicide in a fit of depression... His hypothesis was left without an owner: it was not clear how to prove it or where to test it, and therefore no one took it seriously for a long time. The first response came only thirty years later - almost like in Fermat's era!
The ice broke in 1983, when twenty-seven-year-old German Gerd Faltings announced to the whole world: Mordell's conjecture had been proven! Mathematicians were on their guard, but Faltings was a true German: there were no gaps in his long and complicated proof. It's just that the time has come, the facts and concepts have accumulated - and now one talented algebraist, relying on the results of ten other algebraists, has managed to solve a problem that has stood waiting for the master for sixty years. This is not uncommon in 20th-century mathematics. It is worth recalling the secular continuum problem in set theory, Burnside's two conjectures in group theory, or the Poincaré conjecture in topology. Finally, in number theory, the time has come to harvest the old crops ... Which top will be the next in a series of conquered mathematicians? Will Euler's problem, Riemann's hypothesis, or Fermat's theorem collapse? It is good to!
And now, two years after the revelation of Faltings, another inspired mathematician appeared in Germany. His name was Gerhard Frey, and he claimed something strange: that Fermat's theorem is DERIVED from Taniyama's conjecture! Unfortunately, Frey's style of expressing his thoughts was more reminiscent of the unfortunate Taniyama than his clear compatriot Faltings. In Germany, no one understood Frey, and he went overseas - to the glorious town of Princeton, where, after Einstein, they got used to not such visitors. No wonder Barry Mazur, a versatile topologist, one of the heroes of the recent assault on smooth manifolds, made his nest there. And a student grew up next to Mazur - Ken Ribet, equally experienced in the intricacies of topology and algebra, but still not glorifying himself in any way.
When he first heard Frey's speeches, Ribet decided that this was nonsense and near-science fiction (probably, Weil reacted to Taniyama's revelations in the same way). But Ribet could not forget this "fantasy" and at times returned to it mentally. Six months later, Ribet believed that there was something sensible in Frey's fantasies, and a year later he decided that he himself could almost prove Frey's strange hypothesis. But some "holes" remained, and Ribet decided to confess to his boss Mazur. He listened attentively to the student and calmly replied: “Yes, you have done everything! Here you need to apply the transformation Ф, here - use Lemmas B and K, and everything will take on an impeccable form! So Ribet made a leap from obscurity to immortality, using a catapult in the person of Frey and Mazur. In fairness, all of them - along with the late Taniyama - should be considered proofs of Fermat's Last Theorem.
But here's the problem: they derived their statement from the Taniyama hypothesis, which itself has not been proven! What if she's unfaithful? Mathematicians have long known that “anything follows from a lie”, if Taniyama’s guess is wrong, then Ribet’s impeccable reasoning is worthless! We urgently need to prove (or disprove) Taniyama's conjecture - otherwise someone like Faltings will prove Fermat's theorem in a different way. He will become a hero!
It is unlikely that we will ever know how many young or seasoned algebraists jumped on Fermat's theorem after the success of Faltings or after the victory of Ribet in 1986. All of them tried to work in secret, so that in case of failure they would not be ranked among the community of “dummies”-fermatists. It is known that the most successful of all - Andrew Wiles from Cambridge - felt the taste of victory only at the beginning of 1993. This not so much pleased as frightened Wiles: what if his proof of the Taniyama conjecture showed an error or a gap? Then his scientific reputation perished! You have to carefully write down the proof (but it will be many dozens of pages!) And put it aside for six months or a year, so that later you can re-read it cold-bloodedly and meticulously ... But what if someone publishes their proof during this time? Oh trouble...
Yet Wiles came up with a double way to quickly test his proof. First, you need to trust one of your reliable friends and colleagues and tell him the whole course of reasoning. From the outside, all the mistakes are more visible! Secondly, it is necessary to read a special course on this topic to smart students and graduate students: these smart people will not miss a single lecturer's mistake! Just do not tell them the ultimate goal of the course until the last moment - otherwise the whole world will know about it! And of course, you need to look for such an audience away from Cambridge - it’s better not even in England, but in America ... What could be better than distant Princeton?
Wiles went there in the spring of 1993. His patient friend Niklas Katz, after listening to Wiles' long report, found a number of gaps in it, but all of them were easily corrected. But the Princeton graduate students soon ran away from Wiles's special course, not wanting to follow the whimsical thought of the lecturer, who leads them to no one knows where. After such a (not particularly deep) review of his work, Wiles decided that it was time to reveal a great miracle to the world.
In June 1993, another conference was held in Cambridge, dedicated to the "Iwasawa theory" - a popular section of number theory. Wiles decided to tell his proof of the Taniyama conjecture on it, without announcing the main result until the very end. The report went on for a long time, but successfully, journalists gradually began to flock, who sensed something. Finally, thunder struck: Fermat's theorem is proved! The general rejoicing was not overshadowed by any doubts: everything seems to be clear ... But two months later, Katz, having read the final text of Wiles, noticed another gap in it. A certain transition in reasoning relied on the "Euler system" - but what Wiles built was not such a system!
Wiles checked the bottleneck and realized that he was mistaken here. Even worse: it is not clear how to replace the erroneous reasoning! This was followed by the darkest months of Wiles' life. Previously, he freely synthesized an unprecedented proof from the material at hand. Now he is tied to a narrow and clear task - without the certainty that it has a solution and that he will be able to find it in the foreseeable future. Recently, Frey could not resist the same struggle - and now his name was obscured by the name of the lucky Ribet, although Frey's guess turned out to be correct. And what will happen to MY guess and MY name?
This hard labor lasted exactly one year. In September 1994, Wiles was ready to admit defeat and leave the Taniyama hypothesis to more fortunate successors. Having made such a decision, he began to slowly reread his proof - from beginning to end, listening to the rhythm of reasoning, re-experiencing the pleasure of successful discoveries. Having reached the "damned" place, Wiles, however, did not mentally hear a false note. Was the course of his reasoning still impeccable, and the error arose only in the VERBAL description of the mental image? If there is no “Euler system” here, then what is hidden here?
Suddenly, a simple thought came to me: the "Euler system" does not work where the Iwasawa theory is applicable. Why not apply this theory directly - fortunately, it is close and familiar to Wiles himself? And why did he not try this approach from the very beginning, but got carried away by someone else's vision of the problem? Wiles could no longer remember these details - and it became useless. He carried out the necessary reasoning within the framework of the Iwasawa theory, and everything turned out in half an hour! Thus - with a delay of one year - the last gap in the proof of Taniyama's conjecture was closed. The final text was given to the mercy of a group of reviewers of the most famous mathematical journal, a year later they declared that now there are no errors. Thus, in 1995 last hypothesis Ferma died at the age of three hundred and sixty, turning into a proven theorem that will inevitably enter the textbooks of number theory.
Summing up the three-century fuss around Fermat's theorem, we have to draw a strange conclusion: this heroic epic could not have happened! Indeed, the Pythagorean theorem expresses a simple and important connection between visual natural objects- the length of the segments. But the same cannot be said of Fermat's Theorem. It looks more like a cultural superstructure on a scientific substrate - like reaching the North Pole of the Earth or flying to the moon. Let us recall that both of these feats were sung by writers long before they were accomplished - back in the ancient era, after the appearance of Euclid's Elements, but before the appearance of Diophantus's Arithmetic. So, then there was a public need for intellectual exploits of this kind - at least imaginary! Previously, the Hellenes had had enough of Homer's poems, just as a hundred years before Fermat, the French had had enough of religious passions. But then religious passions subsided - and science stood next to them.
In Russia, such processes began a hundred and fifty years ago, when Turgenev put Yevgeny Bazarov on a par with Yevgeny Onegin. True, the writer Turgenev poorly understood the motives for the actions of the scientist Bazarov and did not dare to sing them, but this was soon done by the scientist Ivan Sechenov and the enlightened journalist Jules Verne. The spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and here comes science fiction first, and then popular science literature (including the magazine "Knowledge is Power").
At the same time, specific scientific theme not at all important to the general public, and not very important even to the hero performers. So, having heard about the achievement of the North Pole by Peary and Cook, Amundsen instantly changed the goal of his already prepared expedition - and soon reached the South Pole, ahead of Scott by one month. Later, Yuri Gagarin's successful circumnavigation of the Earth forced President Kennedy to change the former goal of the American space program to a more expensive but far more impressive one: landing men on the moon.
Even earlier, the insightful Hilbert answered the naive question of students: “The solution of what scientific problem would be most useful now”? - answered with a joke: “Catch a fly on the far side of the moon!” To the perplexed question: “Why is this necessary?” - followed by a clear answer: “Nobody needs THIS! But think of the scientific methods and technical means that we will have to develop to solve such a problem - and what a lot of other beautiful problems we will solve along the way!
This is exactly what happened with Fermat's Theorem. Euler could well have overlooked it.
In this case, some other problem would become the idol of mathematicians - perhaps also from number theory. For example, the problem of Eratosthenes: is there a finite or infinite set of twin primes (such as 11 and 13, 17 and 19, and so on)? Or Euler's problem: is every even number the sum of two prime numbers? Or: is there an algebraic relation between the numbers π and e? These three problems have not yet been solved, although in the 20th century mathematicians have come close to understanding their essence. But this century also gave rise to many new, no less interesting problems, especially at the intersection of mathematics with physics and other branches of natural science.
Back in 1900, Hilbert singled out one of them: to create a complete system of axioms of mathematical physics! A hundred years later, this problem is far from being solved, if only because the arsenal of mathematical means of physics is steadily growing, and not all of them have a rigorous justification. But after 1970, theoretical physics split into two branches. One (classical) since the time of Newton has been modeling and predicting STABLE processes, the other (newborn) is trying to formalize the interaction of UNSTABLE processes and ways to control them. It is clear that these two branches of physics must be axiomatized separately.
The first of them will probably be dealt with in twenty or fifty years ...
And what is missing from the second branch of physics - the one that is in charge of all kinds of evolution (including outlandish fractals and strange attractors, the ecology of biocenoses and Gumilyov's theory of passionarity)? This we are unlikely to understand soon. But the worship of scientists to the new idol has already become a mass phenomenon. Probably, an epic will unfold here, comparable to the three-century biography of Fermat's theorem. Thus, at the intersection of different sciences, new idols are born - similar to religious ones, but more complex and dynamic ...
Apparently, a person cannot remain a person without overthrowing the old idols from time to time and without creating new ones - in pain and with joy! Pierre Fermat was lucky to be at a fateful moment close to the hot spot of the birth of a new idol - and he managed to leave an imprint of his personality on the newborn. One can envy such a fate, and it is not a sin to imitate it.
Sergei Smirnov
"Knowledge is power"