Let's expose! Has Fermat's Last Theorem been proven? For those who don’t press the fields Is there a proof of Fermat’s theorem?
SCIENCE AND TECHNOLOGY NEWS
UDC 51:37;517.958
A.V. Konovko, Ph.D.
Academy of State Fire Service of the Ministry of Emergency Situations of Russia FERMA'S GREAT THEOREM HAS BEEN PROVEN. OR NOT?
For several centuries, it was not possible to prove that the equation xn+yn=zn for n>2 is unsolvable in rational numbers, and therefore in integers. This problem was born under the authorship of the French lawyer Pierre Fermat, who at the same time was professionally engaged in mathematics. Her decision is credited to the American mathematics teacher Andrew Wiles. This recognition lasted from 1993 to 1995.
THE GREAT FERMA"S THEOREM IS PROVED. OR NO?
The dramatic history of Fermat"s last theorem proving is considered. It took almost four hundred years. Pierre Fermat wrote little. He wrote in compressed style. Besides he did not publish his researches. The statement that equation xn+yn=zn is unsolvable on sets of rational numbers and integers if n>2 was attended by Fermat"s commentary that he has found indeed remarkable proving to this statement. The descendants were not reached by this proving. Later this statement was called Fermat "s last theorem. The world best mathematicians broke lance over this theorem without result. In the seventies the French mathematician member of Paris Academy of Sciences Andre Veil laid down new approaches to the solution. In 23 of June, in 1993, at theory of numbers conference in Cambridge, the mathematician of Princeton University Andrew Whiles announced that the Fermat"s last theorem proving is completed. However it was early to triumph.
In 1621, the French writer and lover of mathematics Claude Gaspard Bachet de Meziriak published the Greek treatise "Arithmetic" of Diophantus with a Latin translation and commentary. The luxurious “Arithmetic,” with unusually wide margins, fell into the hands of twenty-year-old Fermat and became his reference book for many years. In its margins he left 48 notes containing the facts he discovered about the properties of numbers. Here, in the margins of “Arithmetic,” Fermat’s great theorem was formulated: “It is impossible to decompose a cube into two cubes or a biquadrate into two biquadrates, or in general a power greater than two into two powers with the same exponent; I found a truly wonderful proof of this, which due to lack of space cannot fit in these fields." By the way, in Latin it looks like this: “Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”
The great French mathematician Pierre Fermat (1601-1665) developed a method for determining areas and volumes and created a new method of tangents and extrema. Along with Descartes, he became the creator of analytical geometry, together with Pascal he stood at the origins of probability theory, in the field of the infinitesimal method general rule differentiation and proved in general view the rule of integration of a power function... But, most importantly, this name is associated with one of the most mysterious and dramatic stories that has ever shocked mathematics - the story of the proof of Fermat's last theorem. Now this theorem is expressed in the form of a simple statement: the equation xn + yn = zn for n>2 is unsolvable in rational numbers, and therefore in integers. By the way, for the case n = 3, the Central Asian mathematician Al-Khojandi tried to prove this theorem in the 10th century, but his proof did not survive.
A native of the south of France, Pierre Fermat received legal education and from 1631 he was an adviser to the parliament of the city of Toulouse (i.e. high court). After a working day within the walls of parliament, he took up mathematics and immediately plunged into a completely different world. Money, prestige, public recognition - none of this mattered to him. Science never became a livelihood for him, did not turn into a craft, always remaining just an exciting game of the mind, understandable only to a few. He carried on his correspondence with them.
Farm never wrote scientific works in our usual understanding. And in his correspondence with friends there is always some challenge, even a kind of provocation, and by no means an academic presentation of the problem and its solution. That’s why many of his letters subsequently came to be called a challenge.
Perhaps this is precisely why he never realized his intention to write a special essay on number theory. Meanwhile, this was his favorite area of mathematics. It was to her that Fermat dedicated the most inspired lines of his letters. “Arithmetic,” he wrote, “has its own field, the theory of integers. This theory was only slightly touched upon by Euclid and was not sufficiently developed by his followers (unless it was contained in those works of Diophantus, which the ravages of time have deprived us of). Arithmeticians, therefore, must develop and renew it."
Why was Fermat himself not afraid of the destructive effects of time? He wrote little and always very concisely. But, most importantly, he did not publish his work. During his lifetime they circulated only in manuscripts. It is not surprising, therefore, that Fermat’s results on number theory have reached us in scattered form. But Bulgakov was probably right: great manuscripts don’t burn! Fermat's work remains. They remained in his letters to friends: the Lyon mathematics teacher Jacques de Billy, the mint employee Bernard Freniquel de Bessy, Marcenny, Descartes, Blaise Pascal... What remained was Diophantus' "Arithmetic" with his comments in the margins, which after Fermat's death were included together with comments by Bachet into the new edition of Diophantus, published by his eldest son Samuel in 1670. Only the evidence itself has not survived.
Two years before his death, Fermat sent his friend Carcavi a letter of testament, which went down in the history of mathematics under the title “Summary of new results in the science of numbers.” In this letter, Fermat proved his famous statement for the case n = 4. But then he was most likely not interested in the statement itself, but in the method of proof he discovered, which Fermat himself called infinite or indefinite descent.
Manuscripts don't burn. But, if not for the dedication of Samuel, who after his father’s death collected all his mathematical sketches and small treatises, and then published them in 1679 under the title “Miscellaneous Mathematical Works,” learned mathematicians would have had to discover and rediscover a lot. But even after their publication, the problems posed by the great mathematician lay motionless for more than seventy years. And this is not surprising. In the form in which they appeared in print, the number-theoretic results of P. Fermat appeared before specialists in the form of serious problems that were not always clear to contemporaries, almost without proof, and indications of internal logical connections between them. Perhaps, in the absence of a coherent, well-thought-out theory lies the answer to the question why Fermat himself never decided to publish a book on number theory. Seventy years later, L. Euler became interested in these works, and this was truly their second birth...
Mathematics paid dearly for Fermat's peculiar manner of presenting his results, as if deliberately omitting their proofs. But, if Fermat claimed that he had proved this or that theorem, then this theorem was subsequently proven. However, there was a hitch with the great theorem.
A mystery always excites the imagination. Entire continents were conquered by the mysterious smile of Gioconda; The theory of relativity, as the key to the mystery of space-time connections, has become the most popular physical theory of the century. And we can safely say that there was no other mathematical problem that was as popular as it was ___93
Scientific and educational problems of civil protection
What is Fermat's theorem? Attempts to prove it led to the creation of an extensive branch of mathematics - the theory of algebraic numbers, but (alas!) the theorem itself remained unproven. In 1908, the German mathematician Wolfskehl bequeathed 100,000 marks to anyone who could prove Fermat's theorem. This was a huge amount for those times! In one moment you could become not only famous, but also get fabulously rich! It is not surprising, therefore, that high school students even in Russia, far from Germany, vying with each other, rushed to prove the great theorem. What can we say about professional mathematicians! But...in vain! After the First World War, money became worthless, and the flow of letters with pseudo-evidence began to dry up, although, of course, it never stopped. They say that the famous German mathematician Edmund Landau prepared printed forms to send to authors of proofs of Fermat’s theorem: “There is an error on page ..., in line ....” (The assistant professor was tasked with finding the error.) There were so many oddities and anecdotes related to the proof of this theorem that one could compile a book out of them. The latest anecdote is A. Marinina’s detective story “Coincidence of Circumstances,” filmed and shown on the country’s television screens in January 2000. In it, our compatriot proves a theorem unproven by all his great predecessors and claims for it Nobel Prize. As is known, the inventor of dynamite ignored mathematicians in his will, so the author of the proof could only claim the Fields Gold Medal, the highest international award approved by mathematicians themselves in 1936.
In the classic work of the outstanding Russian mathematician A.Ya. Khinchin, dedicated to Fermat’s great theorem, provides information on the history of this problem and pays attention to the method that Fermat could have used to prove his theorem. A proof is given for the case n = 4 and short review other important results.
But by the time the detective story was written, and even more so by the time it was filmed, the general proof of the theorem had already been found. On June 23, 1993, at a conference on number theory in Cambridge, Princeton mathematician Andrew Wiles announced that Fermat's Last Theorem had been proven. But not at all as Fermat himself “promised”. The path that Andrew Wiles took was not based on the methods of elementary mathematics. He studied the so-called theory of elliptic curves.
To get an idea of elliptic curves, you need to consider a plane curve defined by a third-degree equation
Y(x,y) = a30X + a21x2y+ ... + a1x+ a2y + a0 = 0. (1)
All such curves are divided into two classes. The first class includes those curves that have sharpening points (such as the semi-cubic parabola y2 = a2-X with the sharpening point (0; 0)), self-intersection points (like the Cartesian sheet x3+y3-3axy = 0, at the point (0; 0)), as well as curves for which the polynomial Dx,y) is represented in the form
f(x^y)=:fl(x^y)■:f2(x,y),
where ^(x,y) and ^(x,y) are polynomials of lower degrees. Curves of this class are called degenerate curves of the third degree. The second class of curves is formed by non-degenerate curves; we will call them elliptic. These may include, for example, the Agnesi Curl (x2 + a2)y - a3 = 0). If the coefficients of the polynomial (1) are rational numbers, then the elliptic curve can be transformed to the so-called canonical form
y2= x3 + ax + b. (2)
In 1955, the Japanese mathematician Y. Taniyama (1927-1958), within the framework of the theory of elliptic curves, managed to formulate a hypothesis that opened the way for the proof of Fermat’s theorem. But neither Taniyama himself nor his colleagues suspected this at the time. For almost twenty years this hypothesis did not attract serious attention and became popular only in the mid-70s. According to the Taniyama conjecture, every elliptic
a curve with rational coefficients is modular. However, so far the formulation of the hypothesis tells little to the meticulous reader. Therefore, some definitions are required.
Each elliptic curve can be associated with an important numerical characteristic - its discriminant. For a curve given in the canonical form (2), the discriminant A is determined by the formula
A = -(4a + 27b2).
Let E be some elliptic curve given by equation (2), where a and b are integers.
For a prime number p, consider the comparison
y2 = x3 + ax + b(mod p), (3)
where a and b are the remainders from dividing the integers a and b by p, and let us denote by np the number of solutions to this comparison. The numbers pr are very useful in studying the question of the solvability of equations of the form (2) in integers: if some pr is equal to zero, then equation (2) has no integer solutions. However, it is possible to calculate numbers only in the rarest cases. (At the same time it is known that р-п|< 2Vp (теоремаХассе)).
Let us consider those prime numbers p that divide the discriminant A of the elliptic curve (2). It can be proven that for such p the polynomial x3 + ax + b can be written in one of two ways:
x3 + ax + b = (x + a)2 (x + ß)(mod P)
x3 + ax + b = (x + y)3 (mod p),
where a, ß, y are some remainders from division by p. If for all primes p dividing the discriminant of the curve, the first of the two indicated possibilities is realized, then the elliptic curve is called semistable.
The prime numbers dividing the discriminant can be combined into what is called an elliptic curve jig. If E is a semistable curve, then its conductor N is given by the formula
where for all prime numbers p > 5 dividing A, the exponent eP is equal to 1. Exponents 82 and 83 are calculated using a special algorithm.
Essentially, this is all that is necessary to understand the essence of the proof. However, Taniyama’s hypothesis contains a complex and, in our case, key concept of modularity. Therefore, let's forget about elliptic curves for a moment and consider the analytic function f (that is, the function that can be represented by a power series) of the complex argument z, given in the upper half-plane.
We denote by H the upper complex half-plane. Let N be a natural number and k be an integer. A modular parabolic form of weight k of level N is an analytic function f(z), defined in the upper half-plane and satisfying the relation
f = (cz + d)kf (z) (5)
for any integers a, b, c, d such that ae - bc = 1 and c is divisible by N. In addition, it is assumed that
lim f (r + it) = 0,
where r is a rational number, and that
The space of modular parabolic forms of weight k of level N is denoted by Sk(N). It can be shown that it has finite dimension.
In what follows, we will be especially interested in modular parabolic forms of weight 2. For small N, the dimension of the space S2(N) is presented in Table. 1. In particular,
Dimensions of the space S2(N)
Table 1
N<10 11 12 13 14 15 16 17 18 19 20 21 22
0 1 0 0 1 1 0 1 0 1 1 1 2
From condition (5) it follows that % + 1) = for each form f e S2(N). Therefore, f is a periodic function. Such a function can be represented as
Let us call a modular parabolic form A^) in S2(N) proper if its coefficients are integers satisfying the relations:
a g ■ a = a g+1 ■ p ■ c Г_1 for a simple p that does not divide the number N; (8)
(ap) for a prime p dividing the number N;
atn = at an, if (t,n) = 1.
Let us now formulate a definition that plays a key role in the proof of Fermat’s theorem. An elliptic curve with rational coefficients and conductor N is called modular if there is such an eigenform
f (z) = ^anq" g S2(N),
that ap = p - pr for almost all prime numbers p. Here n is the number of comparison solutions (3).
It is difficult to believe in the existence of even one such curve. It is quite difficult to imagine that there would be a function A(r) that satisfies the listed strict restrictions (5) and (8), which would be expanded into series (7), the coefficients of which would be associated with practically incomputable numbers Pr. But Taniyama’s bold hypothesis did not at all cast doubt on the fact of their existence, and the empirical material accumulated over time brilliantly confirmed its validity. After two decades of almost complete oblivion, Taniyama's hypothesis received a kind of second wind in the works of the French mathematician, member of the Paris Academy of Sciences Andre Weil.
Born in 1906, A. Weil eventually became one of the founders of a group of mathematicians who acted under the pseudonym N. Bourbaki. Since 1958, A. Weil became a professor at the Princeton Institute for Advanced Study. And the emergence of his interest in abstract algebraic geometry dates back to this same period. In the seventies he turned to elliptic functions and Taniyama's conjecture. The monograph on elliptic functions was translated here in Russia. He is not alone in his hobby. In 1985, the German mathematician Gerhard Frey proposed that if Fermat's theorem is false, that is, if there is a triple of integers a, b, c such that a" + bn = c" (n > 3), then the elliptic curve
y2 = x (x - a")-(x - cn)
cannot be modular, which contradicts Taniyama's conjecture. Frey himself failed to prove this statement, but soon the proof was obtained by the American mathematician Kenneth Ribet. In other words, Ribet showed that Fermat's theorem is a consequence of Taniyama's conjecture.
He formulated and proved the following theorem:
Theorem 1 (Ribet). Let E be an elliptic curve with rational coefficients and having a discriminant
and conductor
Let us assume that E is modular and let
/ (r) = q + 2 aAn e ^ (N)
is the corresponding proper form of level N. We fix a prime number £, and
р:еР =1;- " 8 р
Then there is such a parabolic form
/(g) = 2 dnqn e N)
with integer coefficients such that the differences an - dn are divisible by I for all 1< п<ад.
It is clear that if this theorem is proven for a certain exponent, then it is thereby proven for all exponents divisible by n. Since every integer n > 2 is divisible either by 4 or by an odd prime number, we can therefore limit ourselves to the case when the exponent is either 4 or an odd prime number. For n = 4, an elementary proof of Fermat's theorem was obtained first by Fermat himself, and then by Euler. Thus, it is enough to study the equation
a1 + b1 = c1, (12)
in which the exponent I is an odd prime number.
Now Fermat's theorem can be obtained by simple calculations (2).
Theorem 2. Fermat's last theorem follows from Taniyama's conjecture for semistable elliptic curves.
Proof. Let's assume that Fermat's theorem is false, and let there be a corresponding counterexample (as above, here I is an odd prime). Let us apply Theorem 1 to the elliptic curve
y2 = x (x - ae) (x - c1).
Simple calculations show that the conductor of this curve is given by the formula
Comparing formulas (11) and (13), we see that N = 2. Therefore, by Theorem 1 there is a parabolic form
lying in space 82(2). But by virtue of relation (6), this space is zero. Therefore, dn = 0 for all n. At the same time, a^ = 1. Therefore, the difference ag - dl = 1 is not divisible by I and we arrive at a contradiction. Thus, the theorem is proven.
This theorem provided the key to the proof of Fermat's Last Theorem. And yet the hypothesis itself remained still unproven.
Having announced on June 23, 1993, the proof of the Taniyama conjecture for semistable elliptic curves, which include curves of the form (8), Andrew Wiles was in a hurry. It was too early for mathematicians to celebrate their victory.
The warm summer quickly ended, the rainy autumn was left behind, and winter came. Wiles wrote and rewrote the final version of his proof, but meticulous colleagues found more and more inaccuracies in his work. And so, in early December 1993, a few days before Wiles' manuscript was to go to press, serious gaps in his evidence were again discovered. And then Wiles realized that he couldn’t fix anything in a day or two. This required serious improvement. The publication of the work had to be postponed. Wiles turned to Taylor for help. “Working on the mistakes” took more than a year. The final version of the proof of the Taniyama conjecture, written by Wiles in collaboration with Taylor, was published only in the summer of 1995.
Unlike the hero A. Marinina, Wiles did not apply for the Nobel Prize, but still... he should have been awarded some kind of award. But which one? Wiles was already in his fifties at that time, and Fields’ gold medals are awarded strictly until the age of forty, when the peak of creative activity has not yet passed. And then they decided to establish a special award for Wiles - the silver badge of the Fields Committee. This badge was presented to him at the next congress on mathematics in Berlin.
Of all the problems that can, with greater or lesser probability, take the place of Fermat's last theorem, the problem of the closest packing of balls has the greatest chance. The problem of the densest packing of balls can be formulated as the problem of how to most economically fold oranges into a pyramid. Young mathematicians inherited this task from Johannes Kepler. The problem arose in 1611, when Kepler wrote a short essay “On Hexagonal Snowflakes.” Kepler's interest in the arrangement and self-organization of particles of matter led him to discuss another issue - the densest packing of particles, in which they occupy the smallest volume. If we assume that the particles have the shape of balls, then it is clear that no matter how they are located in space, there will inevitably remain gaps between them, and the question is to reduce the volume of gaps to a minimum. In the work, for example, it is stated (but not proven) that such a shape is a tetrahedron, the coordinate axes inside which determine the basic orthogonality angle of 109°28", and not 90°. This problem is of great importance for particle physics, crystallography and other branches of natural science .
Literature
1. Weil A. Elliptic functions according to Eisenstein and Kronecker. - M., 1978.
2. Soloviev Yu.P. Taniyama's conjecture and Fermat's last theorem // Soros educational journal. - No. 2. - 1998. - P. 78-95.
3. Singh S. Fermat’s Last Theorem. The story of a mystery that has occupied the world's best minds for 358 years / Trans. from English Yu.A. Danilova. M.: MTsNMO. 2000. - 260 p.
4. Mirmovich E.G., Usacheva T.V. Quaternion algebra and three-dimensional rotations // This journal No. 1(1), 2008. - P. 75-80.
It’s unlikely that even a single year in the life of our editorial team passed without it receiving a good dozen proofs of Fermat’s theorem. Now, after the “victory” over her, the flow has subsided, but has not dried up.
Of course, we are not publishing this article in order to dry it completely. And not in my own defense - that, they say, that’s why we kept silent, we ourselves were not yet mature enough to discuss such complex problems.
But if the article really seems complicated, look straight to the end. You will have to feel that passions have subsided temporarily, the science is not over, and soon new proofs of new theorems will be sent to the editors.
It seems that the twentieth century was not in vain. First, people created a second Sun for a moment by exploding a hydrogen bomb. Then they walked on the Moon and finally proved Fermat's famous theorem. Of these three miracles, the first two are on everyone's lips because they caused enormous social consequences. On the contrary, the third miracle looks like just another scientific toy - on a par with the theory of relativity, quantum mechanics and Gödel's theorem on the incompleteness of arithmetic. However, relativity and quanta led physicists to the hydrogen bomb, and the research of mathematicians filled our world with computers. Will this series of miracles continue in the 21st century? Is it possible to trace the connection between the latest scientific toys and revolutions in our everyday life? Does this relationship allow us to make successful predictions? Let's try to understand this using Fermat's theorem as an example.
Let us first note that she was born much later than her natural term. After all, the first special case of Fermat’s theorem is the Pythagorean equation X 2 + Y 2 = Z 2, connecting the lengths of the sides of a right triangle. Having proved this formula twenty-five centuries ago, Pythagoras immediately asked the question: are there many triangles in nature in which both sides and the hypotenuse have a whole 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 relatively prime numbers of different parities. 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 = A 2 - B 2, Y = 2AB, Z = A 2 + B 2) is a solution to the equation X 2 + Y 2 = Z 2 and defines a rectangle with mutual simple side lengths. It is also clear that the number of different triplets of this kind is infinite. But do all solutions to the Pythagorean equation have this form? Pythagoras could neither prove nor refute such a hypothesis and left this problem to his descendants without focusing on it. Who wants to highlight their failures? It seems that after this the problem of integer 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: he was not at all like Pythagoras. He felt like a king in geometry and even beyond it - be it in music, astronomy or politics. The first arithmetic connection between the lengths of the sides of a euphonious 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, a modest researcher at the great Museum, which had long ceased to be the pride of the city crowd, oppose to such successes?
Only one thing: better understanding ancient world numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet know the positional system for writing 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 with the quadratic equation of Pythagoras, and then wondered: does the similar cubic equation X 3 + Y 3 = Z 3 have at least one solution?
Diophantus failed to find such a solution, and his attempt to prove that there are no solutions was also unsuccessful. Therefore, documenting the results of his work in the book “Arithmetic” (this was the world’s first textbook on number theory), Diophantus analyzed the Pythagorean equation in detail, but did not say a word about possible generalizations of this equation. Or it could: after all, it was Diophantus who first proposed notation for powers of integers! But alas: the concept of a “problem book” was alien to Hellenic science and pedagogy, and publishing lists of unsolved problems was considered an indecent activity (only Socrates acted differently). If you can't solve the problem, keep quiet! Diophantus fell silent, and this silence lasted for fourteen centuries - until the advent 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 tireless calculator Kepler tried to guess the relationship between the distances from the Sun to the planets. Pythagoras failed. Kepler achieved success after he learned to integrate polynomials and other simple functions. On the contrary, the visionary Descartes did not like long calculations, but it was he who was the first to present all the points of a plane or space as sets of numbers. This bold model reduces any geometric problem about shapes to some algebraic problem about equations—and vice versa. For example, integer solutions to 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 jungle of integers.
In 1636, a book by Diophantus fell into the hands of a young lawyer from Toulouse, just translated into Latin from the Greek original, which had accidentally survived in some Byzantine archive and was brought to Italy by one of the Roman fugitives at the time of the Turkish devastation. Reading an elegant argument about the Pythagorean equation, Fermat wondered: is it possible to find a solution that consists of three square numbers? There are no small numbers of this kind: it is easy to check by brute force. What about big decisions? Without a computer, Fermat could not carry out a numerical experiment. But he noticed that for each “large” solution of the equation X 4 + Y 4 = Z 4 it is possible to construct a smaller solution. This means that the sum of the fourth powers of two integers is never equal to the same power of the third number! What about the sum of two cubes?
Inspired by the success for degree 4, Fermat tried to modify the "descent method" for degree 3 - and he succeeded. It turned out that it was impossible to make two small cubes from those single cubes into which a large cube with an entire edge length was scattered. The triumphant Fermat made a brief note in the margins of Diophantus's book and sent a letter to Paris with a detailed message about his discovery. But he did not receive an answer - although usually the capital's mathematicians quickly reacted to the latest success of their lonely colleague-rival in Toulouse. What's the matter?
It’s very simple: by the middle of the 17th century, arithmetic went out of fashion. The great successes of 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 had managed to guess the orbits of the planets using pure arithmetic... But alas, this required mathematical analysis. This means that it must be developed - until complete triumph mathematical methods in natural science! But analysis grows out of geometry, while arithmetic remains a field of fun for idle lawyers and other lovers of the eternal science of numbers and figures.
So, Fermat’s arithmetic successes turned out to be untimely and remained unappreciated. He was not upset by this: for the glory of a mathematician, the facts of differential calculus, analytical geometry and probability theory that were revealed to him for the first time were enough. All of these discoveries by 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 18th-century “king of mathematicians” was a champion in all applications of analysis, but he did not neglect arithmetic, 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? Which Hellene could have foreseen that similar series would make it possible to prove the irrationality of the number π?
Such successes forced Euler to carefully re-read Fermat’s surviving manuscripts (fortunately, the son of the great Frenchman managed to publish them). True, the proof of the “grand theorem” for degree 3 has not been preserved, but Euler easily restored it with just one indication of the “descent method”, and immediately tried to transfer this method to the next simple degree - 5.
Not so! In Euler's reasoning, complex numbers appeared, which Fermat managed to overlook (this is the usual lot of discoverers). But factoring complex integers is a delicate matter. Even Euler did not fully understand it and put “Fermat’s problem” aside, rushing to complete his main work - the textbook “Fundamentals of Analysis,” which was supposed to help every talented young man stand on a par with Leibniz and Euler. The publication of the textbook was completed in St. Petersburg in 1770. But Euler never returned 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: Euler’s successor in number theory was the Frenchman Adrien Legendre. IN late XVIII century, he completed the proof of Fermat's theorem for powers 5 - and although he failed for large prime powers, he compiled another textbook on number theory. May his young readers surpass the author just as the readers of “Mathematical Principles of Natural Philosophy” surpassed the great Newton! Legendre was no match for Newton or Euler, but among his readers were two geniuses: 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, capable of discovering or conquering a new world. Many succeeded in this, which is why in the 19th century scientific and technical progress became the main driver of human evolution, 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 everything he wanted. For example, for some reason the ancient problem of trisection of an angle cannot be solved using a compass and ruler. With the help of complex numbers representing points of the plane, Gauss translates this problem into the language of algebra - and obtains a general theory of the feasibility of certain geometric constructions. Thus, at the same time, there appeared a rigorous proof of the impossibility of constructing a regular 7- or 9-gon with a compass and a ruler, and a method for constructing a regular 17-gon, which the wisest geometers of Hellas had never dreamed of.
Of course, such success does not come in vain: we have to invent new concepts that reflect the essence of the matter. Newton introduced three such concepts: fluxion (derivative), fluent (integral) and power series. They were enough to create mathematical analysis and the first scientific model of the physical world, including mechanics and astronomy. Gauss also introduced three new concepts: vector space, field and ring. From them grew a new algebra, which subordinated Greek arithmetic and the theory of numerical functions created by Newton. It still remained to subordinate the logic created by Aristotle to algebra: then it would be possible, using calculations, to prove the deducibility or non-derivability of any scientific statements from a given set of axioms! For example, is Fermat's theorem derived from the axioms of arithmetic, or Euclid's postulate about parallel lines 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 construct the first non-Euclidean geometry, and the first non-commutative algebra (Group Theory) was built by the Frenchman Evariste Galois. And only long after Gauss's death - in 1872 - did the young German Felix Klein realize 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 principle: it is not a royal matter to solve individual problems that do not fit into a clear scientific theory! But Gauss's students, armed with his new algebra and the classical analysis of Newton and Euler, reasoned differently. First, Peter Dirichlet proved Fermat's theorem for the power of 7 using the ring of complex integers generated by the roots of this power of one. Then Ernst Kummer extended the Dirichlet method to ALL prime powers (!) - so it seemed to him in the heat of the moment, and he triumphed. But soon a sobering realization came: the proof is flawless only if every element of the ring can be uniquely decomposed into prime factors! For ordinary integers, this fact was known to Euclid, but only Gauss gave a rigorous proof of it. What about complex integer numbers?
According to the “principle of greatest mischief,” there can and SHOULD be ambiguous factorization! As soon as Kummer learned to calculate the degree of ambiguity using the methods of mathematical analysis, he discovered this dirty trick in the ring for the power of 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 particularly help solve Fermat’s problem: only its natural complexity became clearer.
Throughout the 19th century, this ancient idol demanded more and more victims from its admirers in the form of new complex theories. It is not surprising that by the beginning of the twentieth century, believers became disheartened and rebelled, rejecting their former idol. The word "fermatist" has become a dirty nickname among professional mathematicians. And although a considerable prize was awarded for a complete proof of Fermat’s theorem, its applicants were mostly self-confident ignoramuses. The most powerful mathematicians of that time - Poincaré and Hilbert - pointedly avoided this topic.
In 1900, Hilbert did not include Fermat's theorem in the list of twenty-three most important problems facing mathematics in the twentieth century. True, he included in their series the general problem of 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 thorn will fall out by itself.
This is exactly how the great romantic Henri Poincaré acted. Neglecting many “eternal” problems, all his life he studied SYMMETRIES of certain objects of mathematics or physics: either functions of a complex variable, or trajectories of celestial bodies, or algebraic curves or smooth varieties (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 may be an internal relationship between them, which we are not yet able to comprehend! For example, each of the two-dimensional geometries (Euclidean, Lobachevsky or Riemann) has its own group of symmetries that acts on the plane. But the points of the plane are complex numbers: in this way the action of any geometric group is transferred to the boundless world of complex functions. It is possible and necessary to study the most symmetrical of these functions: AUTOMORPHIC (which are subject to the Euclidean group) and MODULAR (which are subject to the Lobachevsky group)!
There are also elliptic curves on the plane. They are in no way connected with the ellipse, but are given by equations of the form Y 2 = AX 3 + BX 2 + CX and therefore intersect with any 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 we are interested in turns out to be modular, that is, it is related to Lobachevsky geometry...
This is how Poincaré reasoned, seducing the mathematical youth of Europe, but at the beginning of the twentieth century these temptations did not lead to bright theorems or hypotheses. It turned out differently with Hilbert’s call: to study 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 large enough (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 a connection between his hypothesis and Fermat's theorem. If it becomes known that for every degree n > 2 the space of integer solutions to Fermat's equation is finite-dimensional, this will help to prove that there are no such solutions at all! But Mordell did not see any ways to prove his hypothesis - and although he lived a long life, he did not wait for this hypothesis to be transformed into Faltings’ theorem. This happened in 1983 - in a completely different era, after the great successes of the algebraic topology of varieties.
Poincaré created this science as if by accident: he wanted to know what three-dimensional manifolds were. After all, Riemann figured out the structure of all closed surfaces and received a very simple answer! If in a three-dimensional or multidimensional case there is no such answer, you need to come up with a system of algebraic invariants of the variety that determines its geometric structure. It is best if such invariants are elements of some groups - commutative or non-commutative.
Oddly enough, this daring plan of Poincaré was a success: it was carried out from 1950 to 1970 thanks to the efforts of many geometers and algebraists. Until 1950, there was a quiet accumulation of different methods for classifying varieties, and after this date, a critical mass of people and ideas seemed to accumulate and an explosion erupted, comparable to the invention of mathematical analysis in the 17th century. But the analytical revolution stretched over a century and a half, covering creative biographies four generations of mathematicians - from Newton and Leibniz to Fourier and Cauchy. On the contrary, the topological revolution of the twentieth century took place within twenty years - thanks to the large number of its participants. At the same time, a large generation of self-confident young mathematicians was formed, 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. Today, many students of different ages and nationalities, with different abilities and inclinations, circulate between these centers, and everyone wants to become 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 swallow's name 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 organized the first international conference on algebra and number theory in Tokyo. It was apparently easier to do this in Japan, re-educated by the Americans, than in Russia, frozen by Stalin...
Among the guests of honor were two heroes from France: Andre Weil and Jean-Pierre Serre. Here the Japanese were very lucky: Weyl was the recognized head of French algebraists and a member of Bourbaki's group, and the young Serre played a similar role among topologists. In heated discussions with them, the heads of Japanese youth cracked, their brains melted, but 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 expressing himself in English. Then the essence of the matter became clear, but Taniyama was unable to give his hopes a precise formulation. All that Weil could answer the young Japanese was that if he was very lucky in terms of inspiration, then something useful would emerge from his vague hypotheses. But so far there is little hope for this!
Obviously, Weil did not notice the heavenly fire in Taniyama’s gaze. And there was fire: it seemed that for a moment the Japanese was possessed by the indomitable thought of the late Poincaré! Taniyama became convinced that every elliptic curve is generated by modular functions - more precisely, it is “uniformized by a modular form.” Alas, this exact formulation was born much later - in conversations between Taniyama and his friend Shimura. And then Taniyama committed suicide in a fit of depression... His hypothesis was left without 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 hypothesis was proven! The mathematicians were wary, but Faltings was a true German: there were no gaps in his long and complex proof. It’s just that the time has come, facts and concepts have accumulated - and now one talented algebraist, relying on the results of ten other algebraists, managed to solve a problem that had stood waiting for its owner for sixty years. This is not uncommon in twentieth-century mathematics. It is worth recalling the age-old 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 reap the harvest of long-standing crops... Which peak will be the next in the series conquered by mathematicians? Will Euler's problem, the Riemann hypothesis, or Fermat's theorem really collapse? It is good to!
And two years after Faltings’s revelation, another inspired mathematician appeared in Germany. His name was Gerhard Frey, and he claimed something strange: that Fermat’s theorem was DERIVED from the Taniyama conjecture! Unfortunately, in the style of presenting his thoughts, Frey was more reminiscent of the unlucky Taniyama than of his clear-cut compatriot Faltings. In Germany, no one understood Frey, and he went overseas - to the glorious town of Princeton, where, after Einstein, they were accustomed to not such visitors. It’s not for nothing that Barry Mazur, a versatile topologist and one of the heroes of the recent assault on smooth manifolds, has built his nest there. And a student, Ken Ribet, grew up next to Mazur, equally experienced in the intricacies of topology and algebra, but had not yet glorified himself in anything.
When he first heard Frey’s speeches, Ribet decided that it was nonsense and pseudo-science fiction (Weil probably reacted the same way to Taniyama’s revelations). But Ribet could not forget this “fantasy” and from time to time returned to it in his mind. Six months later, Ribet believed that there was something useful in Frey’s fantasies, and a year later he decided that he himself almost knew how to prove Frey’s strange hypothesis. But some “holes” remained, and Ribet decided to confess to his boss Mazur. He listened carefully to the student and calmly replied: “Yes, you’ve got everything done! Here you need to apply the transformation Ф, here you need to use Lemmas B and K, and everything will take on a flawless 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 - together with the late Taniyama - should be considered proof 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 is unfaithful? Mathematicians have long known that “everything 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 attacked Fermat’s theorem after Faltings’ success or after Ribet’s victory in 1986. They all tried to work in secret, so that in case of failure they would not be counted among the community of “dummies”-farmatists. It is known that the luckiest of all, Andrew Wiles from Cambridge, only tasted victory at the beginning of 1993. This did not so much make Wiles happy as it scared him: what if an error or gap was discovered in his proof of the Taniyama conjecture? Then his scientific reputation perished! You need 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 you can then re-read it calmly and meticulously... But what if during this time someone publishes their proof? Oh, trouble...
However, Wiles came up with a double way to quickly check his proof. First, you need to trust one of your reliable colleague friends and tell him the whole line of reasoning. From the outside, all the mistakes are clearer! Secondly, smart students and graduate students need to read a special course on this topic: these smart guys won’t miss a single mistake from the lecturer! Just don’t tell them the final 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 further away from Cambridge - better not even in England, but in America... What could be better than distant Princeton?
Wiles headed there in the spring of 1993. His patient friend Niklas Katz, after listening to Wiles' long report, discovered a number of gaps in it, but all of them were easily corrected. But Princeton graduate students soon ran away from Wiles’ special course, not wanting to follow the whimsical thought of the lecturer, who was leading them to God knows where. After such a (not particularly deep) examination of his work, Wiles decided that it was time to reveal a great miracle to the world.
In June 1993, another conference dedicated to “Iwasawa theory”, a popular branch of number theory, was held in Cambridge. Wiles decided to use it to present his proof of the Taniyama conjecture, without announcing the main result until the very end. The report took a long time, but was successful; journalists gradually began to flock in, sensing something. Finally, thunder struck: Fermat’s theorem was proven! The general rejoicing was not overshadowed by any doubts: everything seemed to be clear... But two months later, Katz, having read Wiles’ final text, noticed another hole in it. A certain transition in reasoning was based on the “Euler system” - but what Wiles built was not such a system!
Wiles checked the bottleneck and realized that he had made a mistake. Worse still: it is unclear how to replace the erroneous reasoning! After this, the darkest months of Wiles' life began. Previously, he freely synthesized an unprecedented proof from available material. Now he is tied to a narrow and clear task - without confidence that it has a solution and that he will be able to find it in the foreseeable time. Recently, Frey could not resist the same struggle - and now his name was obscured by the name of the successful Ribet, although Frey's guess turned out to be correct. What will happen to MY guess and MY name?
This hard labor lasted exactly a year. In September 1994, Wiles was ready to admit defeat and leave the Taniyama hypothesis to more successful successors. Having made this decision, he began to slowly re-read his proof - from beginning to end, listening to the rhythm of reasoning, reliving the pleasure of successful finds. Having reached the “cursed” place, Wiles, however, did not mentally hear a false note. Was his line of reasoning really flawless, and the error arose only during the VERBAL description of the mental image? If there is no “Eulerian system” here, then what is hidden here?
Suddenly a simple thought came to mind: the “Eulerian system” does not work where Iwasawa’s theory is applicable. Why not apply this theory directly - fortunately, Wiles himself is close and familiar with it? And why didn’t he try this approach from the very beginning, but got carried away by someone else’s vision of the problem? Wiles could no longer remember these details - and it was of no use. He carried out the necessary reasoning within the framework of Iwasawa’s theory, and everything worked out in half an hour! Thus, with a delay of one year, the last gap in the proof of the Taniyama conjecture was closed. The final text was left to be torn to pieces by a group of reviewers from a famous mathematical journal; a year later they declared that there were now no errors. Thus, in 1995, Fermat’s last hypothesis died in the three hundred and sixtieth year of his life, turning into a proven theorem that will inevitably be included in number theory textbooks.
Summing up the three-century fuss around Fermat’s theorem, we have to draw a strange conclusion: this heroic epic might not have happened! Indeed, the Pythagorean theorem expresses a simple and important connection between visual natural objects - the lengths of segments. But the same cannot be said about Fermat's theorem. It looks more like a cultural superstructure on a scientific substrate - like reaching the North Pole of the Earth or flying to the Moon. Let us remember that both of these feats were sung by writers long before their accomplishment - back in ancient times, after the appearance of Euclid’s Elements, but before the appearance of Diophantus’ Arithmetic. This means that then a social need arose for intellectual exploits of this kind - at least imaginary ones! Previously, the Hellenes had enough of Homer's poems, just as the French had enough of religious hobbies a hundred years before Fermat. But then religious passions subsided - and science stood next to them.
In Russia, such processes began one and a half hundred 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. A spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and so science fiction appears first, followed by popular science literature (including the magazine “Knowledge is Power”).
At the same time, a specific scientific topic is not at all important for the general public and is not very important even for the performing heroes. 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 flight around the Earth forced President Kennedy to change the previous goal of the American space program to a more expensive, but much more impressive: landing people on the Moon.
Even earlier, the insightful Hilbert answered the naive question of students: “The solution of which scientific problem would be most useful now”? - responded with a joke: “Catch a fly on the far side of the Moon!” To the perplexed question: “Why is this needed?” - came the clear answer: “No one needs THIS! But think about the scientific methods and technical means that we will have to develop to solve such a problem - and what many other beautiful problems we will solve along the way!
This is exactly what happened with Fermat's theorem. Euler could well have missed it.
In this case, some other problem would become the idol of mathematicians - perhaps also from number theory. For example, Eratosthenes' problem: is there a finite or infinite number of twin prime numbers (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 relationship between the numbers π and e? These three problems have still not been solved, although in the twentieth century mathematicians have come noticeably closer to understanding their essence. But this century has also given rise to many new, no less interesting problems, especially at the intersections of mathematics with physics and other branches of natural science.
Back in 1900, Hilbert identified one of them: to create a complete system of axioms of mathematical physics! A hundred years later, this problem is far from being solved, if only because the arsenal of mathematical tools in physics is steadily growing, and not all of them have a strict justification. But after 1970, theoretical physics split into two branches. One (classical) since the time of Newton has been engaged in modeling and forecasting SUSTAINABLE processes, the other (new) 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 strange fractals and strange attractors, the ecology of biocenoses and Gumilyov’s theory of passionarity)? We are unlikely to understand this 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 intersections 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 old idols from time to time and creating new ones - in pain and with joy! Pierre Fermat was lucky to be close to hot spot the birth of a new idol - and he managed to leave the imprint of his personality on the newborn. One can envy such a fate, and it is not a sin to imitate it.
Sergey Smirnov
"Knowledge is power"
In the 17th century, a lawyer and part-time mathematician Pierre Fermat lived in France, who devoted long hours of leisure to his hobby. One winter evening, sitting by the fireplace, he put forward one very curious statement from the field of number theory - it was this that was later called Fermat’s Great Theorem. Perhaps the excitement would not have been so significant in mathematical circles if one event had not happened. The mathematician often spent his evenings studying his favorite book “Arithmetic” by Diophantus of Alexandria (3rd century), while writing down important thoughts in its margins - this rarity was carefully preserved for posterity by his son. So, on the wide margins of this book, Fermat’s hand left the following inscription: “I have a rather striking proof, but it is too large to be placed in the margins.” It was this recording that caused the stunning excitement around the theorem. Mathematicians had no doubt that the great scientist declared that he had proved his own theorem. You are probably asking the question: “Did he really prove it, or was it a banal lie, or maybe there are other versions of why this note, which did not allow mathematicians of subsequent generations to sleep peacefully, ended up in the margins of the book?”
The essence of the Great Theorem
Fermat’s fairly well-known theorem is simple in its essence and lies in the fact that, provided that n is greater than two, a positive number, the equation X n + Y n = Z n will not have solutions of zero type within the framework of natural numbers. This seemingly simple formula masked incredible complexity, and its proof was fought over for three centuries. There is one strange thing - the theorem was late in its birth, since its special case with n = 2 appeared 2200 years ago - this is the no less famous Pythagorean theorem.
It should be noted that the story concerning Fermat’s well-known theorem is very instructive and entertaining, and not only for mathematicians. What is most interesting is that science was not a job for the scientist, but a simple hobby, which in turn gave the Farmer great pleasure. He also constantly kept in touch with a mathematician, and also a friend, and shared ideas, but strangely enough, he did not strive to publish his own works.
Works of the mathematician Farmer
As for the Farmer’s works themselves, they were discovered precisely in the form of ordinary letters. In some places entire pages were missing, and only fragments of correspondence survived. More interesting is the fact that for three centuries scientists have been looking for the theorem that was discovered in the works of Farmer.
But no matter who dared to prove it, the attempts were reduced to “zero.” The famous mathematician Descartes even accused the scientist of boasting, but it all boiled down to just the most common envy. In addition to creating it, the Farmer also proved his own theorem. True, the solution was found for the case where n=4. As for the case for n=3, it was discovered by the mathematician Euler.
How they tried to prove Farmer's theorem
At the very beginning of the 19th century, this theorem continued to exist. Mathematicians found many proofs of theorems that were limited to natural numbers within two hundred.
And in 1909, a fairly large sum was put on the line, equal to one hundred thousand marks of German origin - and all this just to resolve the issue related to this theorem. The prize fund itself was left by a wealthy mathematics lover, Paul Wolfskehl, originally from Germany; by the way, it was he who wanted to “kill himself,” but thanks to such involvement in Fermer’s theorem, he wanted to live. The resulting excitement gave rise to tons of “proofs” that filled German universities, and among mathematicians the nickname “farmist” was born, which was half-contemptuously used to describe any ambitious upstart who was unable to provide clear proofs.
Conjecture of the Japanese mathematician Yutaka Taniyama
Shifts in the history of the Great Theorem were not observed until the mid-20th century, but one interesting event did occur. In 1955, Japanese mathematician Yutaka Taniyama, who was 28 years old, showed the world a statement from a completely different mathematical field - his hypothesis, unlike Fermat’s, was ahead of its time. It says: “Each elliptic curve corresponds to a specific modular shape.” It seems absurd for every mathematician, like the idea that a tree consists of a certain metal! The paradoxical hypothesis, like most other stunning and ingenious discoveries, was not accepted, since they simply had not yet grown up to it. And Yutaka Taniyama committed suicide three years later - an inexplicable act, but probably honor for a true samurai genius was above all else.
The hypothesis was not remembered for a whole decade, but in the seventies it rose to the peak of popularity - it was confirmed by everyone who could understand it, but, like Fermat’s theorem, it remained unproven.
How are Taniyama's conjecture and Fermat's theorem related?
15 years later, a key event occurred in mathematics, and it united the hypothesis of the famous Japanese and Fermat’s theorem. Gerhard Gray stated that when the Taniyama conjecture is proven, then there will be proof of Fermat's theorem. That is, the latter is a consequence of Taniyama’s conjecture, and within a year and a half, Fermat’s theorem was proven by University of California professor Kenneth Ribet.
As time passed, regression was replaced by progress, and science rapidly moved forward, especially in the field of computer technology. Thus, the value of n began to increase more and more.
At the very end of the 20th century, the most powerful computers were located in military laboratories; programming was carried out to output a solution to the well-known Fermat problem. As a consequence of all attempts, it was revealed that this theorem is correct for many values of n, x, y. But, unfortunately, this did not become final proof, since there were no specifics as such.
John Wiles proved Fermat's great theorem
And finally, only at the end of 1994, a mathematician from England, John Wiles, found and demonstrated an exact proof of the controversial Fermer theorem. Then, after many modifications, discussions on this issue came to their logical conclusion.
The refutation was published on more than a hundred pages of one magazine! Moreover, the theorem was proven using a more modern apparatus of higher mathematics. And what is surprising is that at the time when the Farmer wrote his work, such a device did not exist in nature. In a word, the man was recognized as a genius in this field, which no one could argue with. Despite everything that happened, today you can be sure that the presented theorem of the great scientist Farmer is justified and proven, and not a single mathematician with common sense, with which even the most inveterate skeptics of all mankind agree.
The full name of the man after whom the theorem was presented was named Pierre de Fermer. He made contributions to a wide variety of areas of mathematics. But, unfortunately, most of his works were published only after his death.
Envious people claim that the French mathematician Pierre Fermat wrote his name in history with just one phrase. In the margins of the manuscript with the formulation of the famous theorem in 1637, he made a note: “I have found an amazing solution, but there is not enough space to put it here.” 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 understandable even to a schoolchild.
It is based on the Pythagorean theorem, known to everyone: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: x 2 + y 2 = z 2. Fermat argued: the equation for any powers greater than two has no solution in integers.
It would seem simple. Reach out and here is the answer. It's no surprise that 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 the projects " perpetual motion machines"But if serious science has not considered these crazy ideas for a long time, then the work of the "Farmists" is honestly and interestedly studied. And, alas, it finds errors. They say that over more than three centuries a whole mathematical cemetery of solutions to the theorem has formed.
It’s not for nothing that they say: the elbow is close, but you won’t bite. Years, decades, centuries passed, and Fermat’s task seemed increasingly surprising and tempting. Seemingly simple, it turned out to be too tough for the rapidly growing muscle progress. Man had already split the atom, reached the gene, set foot on the moon, but Fermat did not give in, continuing to lure his descendants with false hopes.
However, attempts to overcome the scientific peak were not in vain. The great Euler took the first step by proving the theorem for the fourth degree, then for the third. IN late XIX century, the German Ernst Kummer brought the number of degrees to one hundred. Finally, armed with computers, scientists increased this figure to 100 thousand. But Fermat was talking about any degrees. That was the whole point.
Of course, scientists did not agonize over the problem out of sporting interest. The famous mathematician David Hilbert said that the 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 conquered 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 lost connections between its different sections.
So, the summit has been taken, and science has received the answer,” said Yuri Vishnyakov, scientific secretary of the Department of Mathematics of the Russian Academy of Sciences, Doctor of Technical Sciences, to a RG correspondent. - The theorem has been proven, albeit not in the simplest way, as Fermat himself insisted. And now those who wish can print their own versions.
However, the family of “farmers” is not at all going to accept Wiles’ proof. 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 from another enthusiast appearing on the Internet, “finally putting an end to the long-term epic.”
By the way, just yesterday one of the oldest “fermists” in our country, Vsevolod Yarosh, called the editorial office of “RG”: “And you know that I proved Fermat’s theorem even before Wiles. Moreover, then I found an error in him, which I wrote about to our outstanding mathematician Academician Arnold with a request to publish about this in a scientific journal. Now I am waiting for an answer. I am also corresponding with the French Academy of Sciences about this."
And just now, as reported in a number of media outlets, another enthusiast, former general designer of the Polyot software from Omsk, Doctor of Technical Sciences Alexander Ilyin, with “light grace” revealed the great secret of mathematics. The solution turned out to be so simple and short that it fit on small area newspaper space of one of the central publications.
The editors of RG turned to the country's leading Institute of Mathematics named after. Steklov RAS with a request to evaluate this decision. The scientists were categorical: one cannot comment on the 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 latest proof published. By the way, even a student of the Faculty of Mathematics could easily notice them.
Still, 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 know about such an academy, even among specialists. And when, with the greatest 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 meeting was about to take place with them. historical event. In short, the RG correspondent failed to witness the world sensation.
