Basic research. To whom the fields do not press The essence of the farm theorem

That the 2016 Abel Prize will go to Andrew Wiles for his proof of the Taniyama-Shimura conjecture for semistable elliptic curves and the proof of Fermat's Last Theorem that follows from this conjecture. Currently, the premium is 6 million Norwegian kroner, that is, approximately 50 million rubles. According to Wiles, the award came as a "complete surprise" to him.

Fermat's theorem, proved more than 20 years ago, still attracts the attention of mathematicians. In part, this is due to its formulation, which is understandable even to a schoolboy: to prove that for natural numbers n>2 there are no such triples of non-zero integers that a n + b n = c n . Pierre de Fermat wrote this expression in the margins of Diophantus' Arithmetic, providing it with a wonderful caption: "I have found a truly wonderful proof [of this assertion] for this, but the margins of the book are too narrow for it." Unlike most math tales, this one is real.

The presentation of the award is a great occasion to recall ten entertaining stories related to Fermat's theorem.

1.

Before Andrew Wiles proved Fermat's theorem, it was more properly called a conjecture, that is, Fermat's hypothesis. The fact is that a theorem is, by definition, an already proven statement. However, for some reason, just such a name stuck to this statement.

2.

If we put n = 2 in Fermat's theorem, then such an equation has infinitely many solutions. These solutions are called "Pythagorean triples". They got this name because they correspond to right-angled triangles, the sides of which are expressed by just such sets of numbers. You can generate Pythagorean triples using these three formulas (m 2 - n 2, 2mn, m 2 + n 2). It is necessary to substitute different values of m and n into these formulas, and as a result we will get the triples we need. The main thing here, however, is to make sure that the resulting numbers will be greater than zero - lengths cannot be expressed as negative numbers.

By the way, it is easy to see that if all numbers in a Pythagorean triple are multiplied by some non-zero number, a new Pythagorean triple will be obtained. Therefore, it is reasonable to study triples in which the three numbers in the aggregate do not have a common divisor. The scheme that we have described makes it possible to obtain all such triples - this is by no means a simple result.

3.

On March 1, 1847, at a meeting of the Paris Academy of Sciences, two mathematicians at once - Gabriel Lame and Augustin Cauchy - announced that they were on the verge of proving a remarkable theorem. They ran a race to publish pieces of evidence. Most academics cheered for Lame, because Cauchy was a self-righteous, intolerant religious fanatic (and, of course, an absolutely brilliant part-time mathematician). However, the match was not destined to end - through his friend Joseph Liouville, the German mathematician Ernst Kummer informed the academicians that there was one and the same error in the proofs of Cauchy and Lame.

At school, it is proved that the decomposition of a number into prime factors is unique. Both mathematicians believed that if you look at the decomposition of integers already in the complex case, then this property - uniqueness - will be preserved. However, it is not.

It is noteworthy that if we consider only m + i n, then the decomposition is unique. Such numbers are called Gaussian. But Lame and Cauchy's work required factoring in cyclotomic fields. These are, for example, numbers in which m and n are rational, and i satisfies the property i^k = 1.

4.

Fermat's theorem for n = 3 has a clear geometric sense. Let's imagine that we have many small cubes. Suppose we have collected two large cubes from them. In this case, of course, the sides will be integers. Is it possible to find two such large cubes that, having disassembled them into their component small cubes, we could assemble one large cube from them? Fermat's Theorem says that this can never be done. It's funny that if you ask the same question for three cubes, the answer is yes. For example, there is such a quadruple of numbers, discovered by the wonderful mathematician Srinivas Ramanujan:

3 3 + 4 3 + 5 3 = 6 3

5.

Leonhard Euler was noted in the history of Fermat's theorem. He did not really succeed in proving the statement (or even approaching the proof), but he formulated the hypothesis that the equation

x 4 + y 4 + z 4 = u 4

has no solution in integers. All attempts to find a direct solution to such an equation turned out to be fruitless. It wasn't until 1988 that Nahum Elkies of Harvard managed to find a counterexample. It looks like this:

2 682 440 4 + 15 365 639 4 + 18 796 760 4 = 20 615 673 4 .

Usually this formula is remembered in the context of a numerical experiment. As a rule, in mathematics it looks like this: there is some formula. The mathematician checks this formula in simple cases, convinces himself of the truth and formulates some hypothesis. Then he (although more often some of his graduate students or students) writes a program in order to check that the formula is correct for enough big numbers, which cannot be counted with hands (we are talking about one such experiment with prime numbers). This is not a proof, of course, but an excellent reason to declare a hypothesis. All these constructions are based on the reasonable assumption that if there is a counterexample to some reasonable formula, then we will find it quickly enough.

Euler's conjecture reminds us that life is much more diverse than our fantasies: the first counterexample can be arbitrarily large.

6.

In fact, of course, Andrew Wiles wasn't trying to prove Fermat's Theorem - he was solving a more difficult problem called the Taniyama-Shimura conjecture. There are two remarkable classes of objects in mathematics. The first one is called modular forms and is essentially a function on the Lobachevsky space. These functions do not change during the movements of this very plane. The second is called "elliptic curves" and is the curves given by the equation of the third degree in the complex plane. Both objects are very popular in number theory.

In the 1950s, two talented mathematicians Yutaka Taniyama and Goro Shimura met in the library of the University of Tokyo. At that time, there was no special mathematics at the university: it simply did not have time to recover after the war. As a result, scientists studied using old textbooks and discussed at seminars problems that in Europe and the USA were considered solved and not particularly relevant. It was Taniyama and Shimura who discovered that there is a correspondence between modular forms and elliptic functions.

They tested their conjecture on some simple classes of curves. It turned out that it works. So they suggested that this connection is always there. This is how the Taniyama-Shimura hypothesis appeared, and three years later Taniyama committed suicide. In 1984, the German mathematician Gerhard Frey showed that if Fermat's Theorem is wrong, then the Taniyama-Shimura conjecture is wrong. It followed from this that the one who proved this conjecture would also prove the theorem. And that's exactly what Wiles did - though not in a very general way.

7.

Wiles spent eight years proving the conjecture. And during the check, the reviewers found an error in it, which “killed” most of the proof, nullifying all the years of work. One of the reviewers, by the name of Richard Taylor, undertook to repair the hole with Wiles. While they were working, a message appeared that Elkies, the same one who found a counterexample to Euler's conjecture, also found a counterexample to Fermat's theorem (later it turned out that this was an April Fool's joke). Wiles fell into a depression and did not want to continue - the hole in the evidence could not be closed in any way. Taylor talked Wiles into wrestling for another month.

A miracle happened and by the end of the summer mathematicians managed to make a breakthrough - this is how the works "Modular elliptic curves and Fermat's Last Theorem" by Andrew Wiles (pdf) and "Ring-theoretic properties of some Hecke algebras" by Richard Taylor and Andrew Wiles were born. This was the correct proof. It was published in 1995.

8.

In 1908, the mathematician Paul Wolfskel died in Darmstadt. After himself, he left a will in which he gave the mathematical community 99 years to find a proof of Fermat's Last Theorem. The author of the proof should have received 100 thousand marks (by the way, the author of the counterexample would not have received anything). According to a popular legend, it was love that prompted the Wolfskell mathematicians to make such a gift. Here is how Simon Singh describes the legend in his book Fermat's Last Theorem:

The story begins with Wolfskehl becoming infatuated with a beautiful woman whose identity has never been established. Much to Wolfskel's regret, the mysterious woman rejected him. He fell into such deep despair that he decided to commit suicide. Wolfskel was a passionate man, but not impulsive, and therefore began to work out his death in every detail. He set a date for his suicide and decided to shoot himself in the head with the first strike of the clock at exactly midnight. During the remaining days, Wolfskel decided to put his affairs in order, which were going great, and on the last day he made a will and wrote letters to close friends and relatives.

Wolfskehl worked with such zeal that he finished all his business before midnight and, in order to somehow fill the remaining hours, he went to the library, where he began to look through mathematical journals. He soon came across Kummer's classic paper explaining why Cauchy and Lame had failed. Kummer's work was one of the most significant mathematical publications of its century and was the best reading for a mathematician contemplating suicide. Wolfskel carefully, line by line, followed Kummer's calculations. Unexpectedly, it seemed to Wolfskel that he had discovered a gap: the author made a certain assumption and did not substantiate this step in his reasoning. Wolfskehl wondered if he had really found a serious gap, or if Kummer's assumption was justified. If a gap was found, then there was a chance that Fermat's Last Theorem could be proved much easier than many thought.

Wolfskehl sat down at the table, carefully analyzed the "flawed" part of Kummer's reasoning and began to sketch out a mini-proof, which was supposed to either support Kummer's work or demonstrate the fallacy of his assumption and, as a result, refute all his arguments. By dawn, Wolfskehl had finished his calculations. The bad (mathematically) news was that Kummer's proof had been healed, and Fermat's Last Theorem was still out of reach. But there was good news: the time for suicide had passed, and Wolfskehl was so proud that he had managed to find and fill a gap in the work of the great Ernest Kummer that his despair and sadness dissipated by themselves. Mathematics gave him back the thirst for life.

However, there is an alternative version. According to her, Wolfskel took up mathematics (and, in fact, Fermat's theorem) because of progressive multiple sclerosis, which prevented him from doing what he loved - being a doctor. And he left the money to mathematicians so as not to leave his wife, whom he simply hated by the end of his life.

9.

Attempts to prove Fermat's theorem by elementary methods led to the emergence of a whole class of strange people called "fermatists". They were engaged in the fact that they produced a huge amount of evidence and did not despair at all when they found an error in these proofs.

At the Faculty of Mechanics and Mathematics of Moscow State University there was a legendary character named Dobretsov. He collected certificates from various departments and, using them, penetrated the Mekhmat. This was done solely in order to find the victim. Somehow he came across a young graduate student (the future academician Novikov). He, in his naivety, began to carefully study the stack of papers that Dobretsov slipped him with the words, they say, here is the proof. After another "here's a mistake ..." Dobretsov took the stack and stuffed it into his briefcase. From the second briefcase (yes, he walked around the mekhmat with two briefcases), he took out the second pile, sighed and said: "Well, then let's see option 7 B."

By the way, most of these proofs begin with the phrase "Let's move one of the terms to the right side of the equality and factorize it."

10.

The story about the theorem would be incomplete without the wonderful film "The Mathematician and the Devil".

Amendment

Section 7 of this paper originally stated that Naum Elkies had found a counterexample to Fermat's Theorem, which later turned out to be wrong. This is not true: the message about the counterexample was an April Fool's joke. We apologize for the inaccuracy.

Andrey Konyaev

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, many scientific articles and popular science books appeared that promoted this proof, but none of these works revealed a fundamental mathematical error in it, which crept in not even through the fault of the author, but due to some strange optimism that gripped the minds mathematicians who dealt with this problem and related questions. 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 assertion was reformulated in terms of matching a Diophantine equation of the p-th degree with 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 for elliptic curves: if there are two rational points on a cubic algebraic curve and the line passing through these points intersects this curve at one more point, then the latter is also a rational point. 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 obtained in this way exist (the second unconfirmed assumption). Thirdly, since it is proved by other methods that the constructed concrete curve is non-modular, it means that it does not exist. 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 in the second step of the proof process, when it is assumed that a hypothetical solution to Fermat's equation is also a solution to a third-degree algebraic equation describing an elliptic curve of a known form. In itself, such an assumption would be justified if the indicated curve were indeed 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". AT this case an integer solution of Fermat's equation is compared (apparently, presumably uniquely) with a fictitious, nonexistent 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 Fermat's 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 the equation, according to appearance very similar to the standard form:

Y 2 \u003d - (X - A) (X + B),

where A and B are no longer variables, but concrete numbers determined by the above 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 trust, 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.

Recently, 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 substitute 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 - Unified Science Magazine(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' ERRONEOUS PROOF OF FERMAT'S GREAT THEOREM // Fundamental Research. - 2008. - No. 3. - P. 13-16;
URL: http://fundamental-research.ru/ru/article/view?id=2763 (date of access: 03.03.2020). We bring to your attention the journals published by the publishing house "Academy of Natural History"

So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in its essence and understandable to any person with 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 5th grade high school, but the proof is not even any professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved for so long. 2. What does it consist of?

Let's start with Pythagorean pants The wording is really simple - at first glance. As we know from childhood, "Pythagorean pants are equal on all sides." The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle the square built on the hypotenuse is equal to the sum of the squares built on the legs.

In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triples satisfying the equation x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They must have tried looking for threes or more. high 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 studied the 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 contain it."

Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.

After Fermat, 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 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 new method proof of. 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 is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.

In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.

While the hype continued in the press, serious work began to verify the evidence. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.

It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.

“...half a minute after the start of the festive dinner on the occasion of her birthday, I gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?

This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.

A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!

Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ...

Judging by the popularity of the query "Fermat's theorem - short proof, this mathematical problem a lot of people are really interested. This theorem was first stated by Pierre de Fermat in 1637 on the edge of a copy of the Arithmetic, where he claimed that he had a solution that was too large to fit on the edge.

The first successful proof was published in 1995, the complete proof of Fermat's Theorem by Andrew Wiles. It has been described as "staggering progress" and led Wiles to receive the Abel Prize in 2016. Although described relatively briefly, the proof of Fermat's theorem also proved much of the modularity theorem and opened up new approaches to numerous other problems and effective methods the rise of modularity. These accomplishments have advanced mathematics 100 years into the future. The proof of Fermat's little theorem today is not something out of the ordinary.

The unresolved problem stimulated the development of algebraic number theory in the 19th century and the search for a proof of the modularity theorem in the 20th century. This is one of the most notable theorems in the history of mathematics, and until the full proof of Fermat's Last Theorem by division, it was in the Guinness Book of Records as "the most difficult mathematical problem", one of the features of which is that it has the largest number of unsuccessful proofs.

History reference

The Pythagorean equation x 2 + y 2 = z 2 has an infinite number of positive integer solutions for x, y and z. These solutions are known as Pythagorean trinities. Around 1637, Fermat wrote on the edge of the book that the more general equation a n + b n = c n has no solutions in natural numbers, if n is an integer greater than 2. Although Fermat himself claimed to have a solution to his problem, he left no details about its proof. The elementary proof of Fermat's theorem, claimed by its creator, was rather his boastful invention. The book of the great French mathematician was discovered 30 years after his death. This equation, called Fermat's Last Theorem, remained unsolved in mathematics for three and a half centuries.

The theorem eventually became one of the most notable unsolved problems in mathematics. Attempts to prove this caused a significant development in number theory, and over time Fermat's last theorem became known as an unsolved problem in mathematics.

A Brief History of the Evidence

If n = 4, as proved by Fermat himself, it suffices to prove the theorem for indices n that are prime numbers. Over the next two centuries (1637-1839) the conjecture was only proven for the primes 3, 5 and 7, although Sophie Germain updated and proved an approach that applied to the whole class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, resulting in irregular prime numbers analyzed individually. Based on Kummer's work and using sophisticated computer research, other mathematicians were able to extend the solution of the theorem, with the goal of covering all the main exponents up to four million, but the proof for all exponents was still not available (meaning that mathematicians usually considered the solution of the theorem impossible, extremely difficult, or unattainable with current knowledge).

The work of Shimura and Taniyama

In 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected that there was a connection between elliptic curves and modular forms, two very different branches of mathematics. Known at the time as the Taniyama-Shimura-Weil conjecture and (ultimately) as the modularity theorem, it existed on its own, with no apparent connection to Fermat's last theorem. It itself was widely regarded as an important mathematical theorem, but it was considered (like Fermat's theorem) impossible to prove. At the same time, the proof of Fermat's Last Theorem (by dividing and applying complex mathematical formulas) was not completed until half a century later.

In 1984, Gerhard Frey noticed an obvious connection between these two previously unrelated and unresolved problems. A complete confirmation that the two theorems were closely related was published in 1986 by Ken Ribet, who based on a partial proof by Jean-Pierre Serra, who proved all but one part, known as the "epsilon hypothesis". Simply put, these works by Frey, Serra, and Ribe showed that if the modularity theorem could be proved, at least for a semistable class of elliptic curves, then the proof of Fermat's last theorem would sooner or later be discovered as well. Any solution that can contradict Fermat's last theorem can also be used to contradict the modularity theorem. Therefore, if the modularity theorem turned out to be true, then by definition there cannot be a solution that contradicts Fermat's last theorem, which means that it should have been proved soon.

Although both theorems were difficult problems in mathematics, considered unsolvable, the work of the two Japanese was the first suggestion of how Fermat's last theorem could be extended and proven for all numbers, not just some. Important for the researchers who chose the research topic was the fact that, unlike Fermat's last theorem, the modularity theorem was the main active area of research for which the proof was developed, and not just a historical oddity, so the time spent on its work could be justified from a professional point of view. However, the general consensus was that solving the Taniyama-Shimura hypothesis proved to be inexpedient.

Fermat's Last Theorem: Wiles' proof

After learning that Ribet had proven Frey's theory correct, English mathematician Andrew Wiles, who had been interested in Fermat's Last Theorem since childhood and had experience with elliptic curves and adjacent domains, decided to try to prove the Taniyama-Shimura Conjecture as a way to prove Fermat's Last Theorem. In 1993, six years after announcing his goal, while secretly working on the problem of solving the theorem, Wiles managed to prove a related conjecture, which in turn would help him prove Fermat's last theorem. Wiles' document was enormous in size and scope.

A flaw was discovered in one part of his original paper during peer review and required another year of collaboration with Richard Taylor to jointly solve the theorem. As a result, Wiles' final proof of Fermat's Last Theorem was not long in coming. In 1995, it was published on a much smaller scale than Wiles's previous mathematical work, illustrating that he was not mistaken in his previous conclusions about the possibility of proving the theorem. Wiles' achievement was widely publicized in the popular press and popularized in books and television programs. The remaining parts of the Taniyama-Shimura-Weil conjecture, which have now been proven and are known as the modularity theorem, were subsequently proved by other mathematicians who built on Wiles' work between 1996 and 2001. For his achievement, Wiles has been honored and received numerous awards, including the 2016 Abel Prize.

Wiles' proof of Fermat's last theorem is a special case of solving the modularity theorem for elliptic curves. However, this is the most famous case of such a large-scale mathematical operation. Along with solving Ribe's theorem, the British mathematician also obtained a proof of Fermat's last theorem. Fermat's Last Theorem and Modularity Theorem were almost universally considered unprovable by modern mathematicians, but Andrew Wiles was able to prove everything scientific world that even pundits can be wrong.

Wiles first announced his discovery on Wednesday 23 June 1993 at a Cambridge lecture titled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993, it was found that his calculations contained an error. A year later, on September 19, 1994, in what he would call "the most important moment of his working life," Wiles stumbled upon a revelation that allowed him to fix the solution to the problem to the point where it could satisfy the mathematical community.

Job Description

Andrew Wiles' proof of Fermat's Theorem uses many methods from algebraic geometry and number theory, and has many ramifications in these areas of mathematics. He also uses the standard constructions of modern algebraic geometry, such as the category of schemes and the Iwasawa theory, as well as other 20th-century methods that were not available to Pierre de Fermat.

The two papers containing the evidence are 129 pages long and were written over the course of seven years. John Coates described this discovery as one of the greatest achievements of number theory, and John Conway called it the major mathematical achievement of the 20th century. Wiles, in order to prove Fermat's last theorem by proving the modularity theorem for the special case of semistable elliptic curves, developed powerful methods for lifting modularity and opened up new approaches to numerous other problems. For solving Fermat's last theorem, he was knighted and received other awards. When it became known that Wiles had won the Abel Prize, the Norwegian Academy of Sciences described his achievement as "a delightful and elementary proof of Fermat's Last Theorem".

How it was

One of the people who reviewed Wiles' original manuscript with the solution to the theorem was Nick Katz. In the course of his review, he asked the Briton a number of clarifying questions that led Wiles to admit that his work clearly contains a gap. In one critical part of the proof, an error was made that gave an estimate for the order of a particular group: the Euler system used to extend the Kolyvagin and Flach method was incomplete. The mistake, however, did not make his work useless - every part of Wiles's work was very significant and innovative in itself, as were many of the developments and methods that he created in the course of his work and which affected only one part of the manuscript. However, this original work, published in 1993, did not really have a proof of Fermat's Last Theorem.

Wiles spent almost a year trying to rediscover a solution to the theorem, first alone and then in collaboration with his former student Richard Taylor, but all seemed to be in vain. By the end of 1993, rumors had circulated that Wiles's proof had failed in testing, but how serious this failure was was not known. Mathematicians began to put pressure on Wiles to reveal the details of his work, whether it was done or not, so that the wider community of mathematicians could explore and use whatever he was able to achieve. Instead of quickly correcting his mistake, Wiles only discovered additional difficult aspects in the proof of Fermat's Last Theorem, and finally realized how difficult it was.

Wiles states that on the morning of September 19, 1994, he was on the verge of giving up and giving up, and was almost resigned to failing. He was ready to publish his unfinished work so that others could build on it and find where he was wrong. The English mathematician decided to give himself one last chance and analyzed the theorem for the last time to try to understand the main reasons why his approach did not work, when he suddenly realized that the Kolyvagin-Flac approach would not work until he connected more and more to the proof process Iwasawa's theory by making it work.

On October 6, Wiles asked three colleagues (including Fultins) to review him new job, and on October 24, 1994, he submitted two manuscripts - "Modular elliptic curves and Fermat's last theorem" and "Theoretical properties of the ring of some Hecke algebras", the second of which Wiles wrote jointly with Taylor and proved that certain conditions necessary for to justify the corrected step in the main article.

These two papers were reviewed and finally published as a full text edition in the May 1995 Annals of Mathematics. Andrew's new calculations were widely analyzed and eventually accepted by the scientific community. In these works, the modularity theorem for semistable elliptic curves was established - the last step towards proving Fermat's Last Theorem, 358 years after it was created.

History of the Great Problem

Solving this theorem has been considered the biggest problem in mathematics for many centuries. In 1816 and in 1850 the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857 the Academy awarded 3,000 francs and gold medal Kummer for his research on ideal numbers, although he did not apply for the prize. Another prize was offered to him in 1883 by the Brussels Academy.

Wolfskel Prize

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks (a large amount for the time) to the Göttingen Academy of Sciences to be the prize for the complete proof of Fermat's Last Theorem. On June 27, 1908, the Academy published nine award rules. Among other things, these rules required the proof to be published in a peer-reviewed journal. The prize was to be awarded only two years after publication. The competition was due to expire on September 13, 2007 - about a century after it began. On June 27, 1997, Wiles received Wolfschel's prize money and then another $50,000. In March 2016, he received 600,000 euros from the Norwegian government as part of the Abel Prize for "an amazing proof of Fermat's last theorem with the help of the modularity conjecture for semistable elliptic curves, discovering new era in number theory. It was the world triumph of the humble Englishman.

Prior to Wiles's proof, Fermat's Theorem, as mentioned earlier, was considered absolutely unsolvable for centuries. Thousands of False Evidence in different time were presented to the Wolfskell committee, amounting to approximately 10 feet (3 meters) of correspondence. Only in the first year of the existence of the prize (1907-1908) 621 applications were submitted claiming to solve the theorem, although by the 1970s their number had decreased to about 3-4 applications per month. According to F. Schlichting, Wolfschel's reviewer, most of the evidence was based on elementary methods taught in schools and was often presented as "people with a technical background but a failed career". According to the historian of mathematics Howard Aves, Fermat's last theorem set a kind of record - it is the theorem with the most incorrect proofs.

Fermat's laurels went to the Japanese

As discussed earlier, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama discovered a possible connection between two apparently completely different branches of mathematics - elliptic curves and modular forms. The resulting modularity theorem (then known as the Taniyama-Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

The theory was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence to support the Japanese conclusions. As a result, the hypothesis has often been referred to as the Taniyama-Shimura-Weil hypothesis. It became part of the Langlands program, which is a list of important hypotheses that need to be proven in the future.

Even after serious scrutiny, the conjecture has been recognized by modern mathematicians as extremely difficult, or perhaps inaccessible to proof. Now it is this theorem that is waiting for its Andrew Wiles, who could surprise the whole world with its solution.

Fermat's Theorem: Perelman's proof

Despite the common myth, the Russian mathematician Grigory Perelman, for all his genius, has nothing to do with Fermat's theorem. That, however, does not detract from his numerous merits to the scientific community.