FERMAT GREAT THEOREM - the statement of Pierre Fermat (a French lawyer and part-time mathematician) that the Diophantine equation X n + Y n = Z n , with an exponent n>2, where n = an integer, has no solutions in positive integers . Author's text: "It is impossible to decompose a cube into two cubes, or a bi-square into two bi-squares, or in general a power greater than two into two powers with the same exponent."
"Fermat and his theorem", Amadeo Modigliani, 1920
Pierre came up with this theorem on March 29, 1636. And after some 29 years, he died. But that's where it all started. After all, a wealthy German mathematician by the name of Wolfskel bequeathed one hundred thousand marks to the one who presents the complete proof of Fermat's theorem! But the excitement around the theorem was connected not only with this, but also with professional mathematical excitement. Fermat himself hinted to the mathematical community that he knew the proof - shortly before his death, in 1665, he left the following entry in the margins of the book Diophantus of Alexandria "Arithmetic": "I have a very amazing proof, but it is too large to be placed on fields."
It was this hint (plus, of course, a cash prize) that made mathematicians unsuccessfully spend their best years searching for proof (according to American scientists, professional mathematicians alone spent 543 years on this in total).
At some point (in 1901), work on Fermat's theorem acquired the dubious fame of "work akin to the search for a perpetual motion machine" (there was even a derogatory term - "fermatists"). And suddenly, on June 23, 1993, at a mathematical conference on number theory in Cambridge, English professor of mathematics from Princeton University (New Jersey, USA) Andrew Wiles announced that he had finally proved Fermat!
The proof, however, was not only complicated, but also obviously erroneous, as Wiles was pointed out by his colleagues. But Professor Wiles dreamed of proving the theorem all his life, so it is not surprising that in May 1994 he presented a new, improved version of the proof to the scientific community. There was no harmony, beauty in it, and it was still very complicated - the fact that mathematicians have been analyzing this proof for a whole year (!) To understand whether it is not erroneous, speaks for itself!
But in the end, Wiles' proof was found to be correct. But mathematicians did not forgive Pierre Fermat for his very hint in Arithmetic, and, in fact, they began to consider him a liar. In fact, the first person to question Fermat's moral integrity was Andrew Wiles himself, who remarked that "Fermat could not have had such proof. This is twentieth-century proof." Then, among other scientists, the opinion became stronger that Fermat "could not prove his theorem in another way, and Fermat could not prove it in the way that Wiles went, for objective reasons."
In fact, Fermat, of course, could prove it, and a little later this proof will be recreated by the analysts of the New Analytical Encyclopedia. But - what are these "objective reasons"?
In fact, there is only one such reason: in those years when Fermat lived, Taniyama's conjecture could not appear, on which Andrew Wiles built his proof, because the modular functions that Taniyama's conjecture operates on were discovered only at the end of the 19th century.
How did Wiles himself prove the theorem? The question is not idle - this is important for understanding how Fermat himself could prove his theorem. Wiles built his proof on the proof of Taniyama's conjecture put forward in 1955 by the 28-year-old Japanese mathematician Yutaka Taniyama.
The conjecture sounds like this: "every elliptic curve corresponds to a certain modular form." Elliptic curves, known for a long time, have a two-dimensional form (located on a plane), while modular functions have a four-dimensional form. That is, Taniyama's hypothesis combined completely different concepts - simple flat curves and unimaginable four-dimensional forms. The very fact of connecting different-dimensional figures in the hypothesis seemed absurd to scientists, which is why in 1955 it was not given any importance.
However, in the fall of 1984, the "Taniyama hypothesis" was suddenly remembered again, and not only remembered, but its possible proof was connected with the proof of Fermat's theorem! This was done by Saarbrücken mathematician Gerhard Frey, who told the scientific community that "if anyone could prove Taniyama's conjecture, then Fermat's Last Theorem would be proved."
What did Frey do? He converted Fermat's equation to a cubic one, then drew attention to the fact that an elliptic curve obtained by converting Fermat's equation to a cubic one cannot be modular. However, Taniyama's conjecture stated that any elliptic curve could be modular! Accordingly, an elliptic curve constructed from Fermat's equation cannot exist, which means there cannot be entire solutions and Fermat's theorem, which means it is true. Well, in 1993, Andrew Wiles simply proved Taniyama's conjecture, and hence Fermat's theorem.
However, Fermat's theorem can be proved much more simply, on the basis of the same multidimensionality that both Taniyama and Frey operated on.
To begin with, let's pay attention to the condition stipulated by Pierre Fermat himself - n>2. Why was this condition necessary? Yes, only for the fact that for n=2 the ordinary Pythagorean theorem X 2 +Y 2 =Z 2 becomes a special case of Fermat's theorem, which has an infinite number of integer solutions - 3,4,5; 5,12,13; 7.24.25; 8,15,17; 12,16,20; 51,140,149 and so on. Thus, the Pythagorean theorem is an exception to Fermat's theorem.
But why exactly in the case of n=2 does such an exception occur? Everything falls into place if you see the relationship between the degree (n=2) and the dimension of the figure itself. The Pythagorean triangle is a two-dimensional figure. Not surprisingly, Z (that is, the hypotenuse) can be expressed in terms of legs (X and Y), which can be integers. The size of the angle (90) makes it possible to consider the hypotenuse as a vector, and the legs are vectors located on the axes and coming from the origin. Accordingly, it is possible to express a two-dimensional vector that does not lie on any of the axes in terms of the vectors that lie on them.
Now, if we go to the third dimension, and hence to n=3, in order to express a three-dimensional vector, there will not be enough information about two vectors, and therefore it will be possible to express Z in Fermat's equation in at least three terms (three vectors lying, respectively, on the three axes of the coordinate system).
If n=4, then there should be 4 terms, if n=5, then there should be 5 terms, and so on. In this case, there will be more than enough whole solutions. For example, 3 3 +4 3 +5 3 =6 3 and so on (you can choose other examples for n=3, n=4 and so on).
What follows from all this? It follows from this that Fermat's theorem does indeed have no entire solutions for n>2 - but only because the equation itself is incorrect! With the same success, one could try to express the volume of a parallelepiped in terms of the lengths of its two edges - of course, this is impossible (whole solutions will never be found), but only because to find the volume of a parallelepiped, you need to know the lengths of all three of its edges.
When the famous mathematician David Gilbert was asked what is the most important task for science now, he answered "to catch a fly on the far side of the moon." To the reasonable question "Who needs it?" he answered like this: "No one needs it. But think about how many important and complex tasks you need to solve in order to accomplish this."
In other words, Fermat (a lawyer in the first place!) played a witty legal joke on the entire mathematical world, based on an incorrect formulation of the problem. He, in fact, suggested that mathematicians find an answer why a fly cannot live on the other side of the Moon, and in the margins of Arithmetic he only wanted to write that there is simply no air on the Moon, i.e. there can be no integer solutions of his theorem for n>2 only because each value of n must correspond to a certain number of terms on the left side of his equation.
But was it just a joke? Not at all. Fermat's genius lies precisely in the fact that he was actually the first to see the relationship between the degree and the dimension of a mathematical figure - that is, which is absolutely equivalent, the number of terms on the left side of the equation. The meaning of his famous theorem was precisely to not only push the mathematical world on the idea of this relationship, but also to initiate the proof of the existence of this relationship - intuitively understandable, but mathematically not yet substantiated.
Fermat, like no one else, understood that establishing a relationship between seemingly different objects is extremely fruitful not only in mathematics, but also in any science. Such a relationship points to some deep principle underlying both objects and allowing a deeper understanding of them.
For example, initially physicists considered electricity and magnetism as completely unrelated phenomena, and in the 19th century, theorists and experimenters realized that electricity and magnetism were closely related. The result was a deeper understanding of both electricity and magnetism. Electric currents generate magnetic fields, and magnets can induce electricity in conductors that are close to the magnets. This led to the invention of dynamos and electric motors. Eventually it was discovered that light is the result of coordinated harmonic oscillations of magnetic and electric fields.
The mathematics of Fermat's time consisted of islands of knowledge in a sea of ignorance. Geometers studied shapes on one island, and mathematicians studied probability and chance on the other island. The language of geometry was very different from the language of probability theory, and algebraic terminology was alien to those who spoke only about statistics. Unfortunately, mathematics of our time consists of approximately the same islands.
Farm was the first to realize that all these islands are interconnected. And his famous theorem - Fermat's GREAT THEOREM - is an excellent confirmation of this.
In the 17th century, a lawyer and part-time mathematician Pierre Fermat lived in France, who gave his hobby long hours of leisure. One winter evening, sitting by the fireplace, he put forward one most curious statement from the field of number theory - it was this that was later called Fermat's Great or Great Theorem. Perhaps the excitement would not have been so significant in mathematical circles if one event had not happened. The mathematician often spent evenings studying the favorite book of Diophantus of Alexandria "Arithmetic" (3rd century), while writing down important thoughts in its margins - this rarity was carefully preserved for posterity by his son. So, in the wide margins of this book, Fermat's hand had left this inscription: "I have a rather striking proof, but it is too large to be placed in the margins." It was this entry that caused the overwhelming excitement around the theorem. There was no doubt among mathematicians that the great scientist declared that he had proved his own theorem. You are probably wondering: “Did he really prove it, or was it a banal lie, or maybe there are other versions, why did this entry, which did not allow mathematicians of subsequent generations to sleep peacefully, end up on the margins of the book?”.
The essence of the Great Theorem
The rather well-known Fermat's theorem is simple in its essence and consists in the fact that, provided that n is greater than two, a positive number, the equation X n + Y n \u003d Z n will not have solutions of zero type within the framework of natural numbers. Incredible complexity was masked in this seemingly simple formula, and it took three centuries to prove it. There is one oddity - the theorem was late with its birth, since its special case for n = 2 appeared 2200 years ago - this is the no less famous Pythagorean theorem.
It should be noted that the story concerning the well-known Fermat's 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 part-time, also a friend, shared ideas, but oddly enough, he did not seek to publish his own work.
Proceedings of the mathematician Farmer
As for the works of Farmer themselves, they were found precisely in the form of ordinary letters. In some places there were no whole pages, and only fragments of correspondence have been preserved. More interesting is the fact that for three centuries scientists have been looking for the theorem that was discovered in the writings of Fermer.
But whoever did not dare to prove it, attempts were reduced to "zero". The famous mathematician Descartes even accused the scientist of boasting, but it all boiled down to the most ordinary envy. In addition to creating, Farmer also proved his own theorem. True, the solution was found for the case where n=4. As for the case for n=3, the mathematician Euler identified it.
How did they try to prove Fermer's theorem
At the very beginning of the 19th century, this theorem continued to exist. Mathematicians have found many proofs of theorems that were limited to natural numbers within two hundred.
And in 1909, a rather large amount was put on the line, equal to one hundred thousand marks of German origin - and all this just to solve the problem associated with this theorem. The fund of the prize category itself was left by a wealthy math lover Paul Wolfskell, originally from Germany, by the way, it was he who wanted to "lay hands on himself", but thanks to such involvement in Fermer's theorem, he wanted to live. The resulting excitement gave rise to tons of "proof" that flooded the German universities, and in the circle of mathematicians, the nickname "fermist" was born, which was used semi-contemptuously to call any ambitious upstart who failed to provide clear evidence.
Hypothesis of the Japanese mathematician Yutaka Taniyama
There were no shifts in the history of the Great Theorem until the middle of the 20th century, but one interesting event did occur. In 1955, the Japanese mathematician Yutaka Taniyama, who was 28 years old, revealed to the world a statement from a completely different mathematical field - his hypothesis, unlike Fermat, was ahead of its time. It says: "For every elliptic curve there is a corresponding modular form." It seems to be an absurdity for every mathematician, like that a tree consists of a certain metal! The paradoxical hypothesis, like most other stunning and ingenious discoveries, was not accepted, because they simply had not grown up to it yet. And Yutaka Taniyama committed suicide three years later - an inexplicable act, but, probably, the honor for a true samurai genius was above all.
For a whole decade, the conjecture was not remembered, 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 Taniyama's Conjecture and Fermat's Theorem Are Related
Fifteen years later, a key event occurred in mathematics, and it combined the famous Japanese conjecture and Fermat's theorem. Gerhard Gray stated that when the Taniyama conjecture is proved, then the proofs of Fermat's theorem will be found. That is, the latter is a consequence of the Taniyama hypothesis, and a year and a half later, Fermat's theorem was proved by a professor at the University of California, Kenneth Ribet.
Time passed, regression was replaced by progress, and science was rapidly moving 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 in military laboratories, programming was carried out to derive 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 the final proof, since there was 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 improvements, discussions on this subject came to their logical conclusion.
The rebuttal was posted on more than a hundred pages of one magazine! Moreover, the theorem was proved on a more modern apparatus of higher mathematics. And surprisingly, at the time when the Farmer wrote his work, such an apparatus 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 no mathematician with common sense will start disputes on this topic, which even the most inveterate skeptics of all mankind agree with.
The full name of the person after whom the presented theorem was named was 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.
The Grand Theorem Farm Singh Simon
"Has Fermat's Last Theorem been proven?"
It was only the first step towards proving the Taniyama-Shimura conjecture, but the strategy chosen by Wiles was a brilliant mathematical breakthrough, a result that deserved to be published. But due to the vow of silence imposed by Wiles on himself, he could not tell the rest of the world about the result and had no idea who else could make such a significant breakthrough.
Wiles recalls his philosophical attitude towards any potential challenger: “No one wants to spend years proving something and find that someone else managed to find the proof a few weeks earlier. But, oddly enough, since I was trying to solve a problem that was essentially considered insoluble, I was not very afraid of my opponents. I just didn't expect myself or anyone else to come up with an idea that would lead to a proof."
On March 8, 1988, Wiles was shocked to see front-page headlines in large print that read: "Fermat's Last Theorem Proven." The Washington Post and the New York Times reported that 38-year-old Yoichi Miyaoka of Tokyo Metropolitan University had solved the world's most difficult mathematical problem. So far, Miyaoka has not yet published his proof, but he outlined its course at a seminar at the Max Planck Institute for Mathematics in Bonn. Don Zagier, who attended Miyaoka's report, expressed the optimism of the mathematical community in the following words: “The proof presented by Miyaoka is extremely interesting, and some mathematicians believe that it will turn out to be correct with a high probability. There is no certainty yet, but so far the evidence looks very encouraging.”
Speaking at a seminar in Bonn, Miyaoka spoke about his approach to solving the problem, which he considered from a completely different, algebro-geometric, point of view. Over the past decades, geometers have achieved a deep and subtle understanding of mathematical objects, in particular, the properties of surfaces. In the 1970s, the Russian mathematician S. Arakelov tried to establish parallels between problems in algebraic geometry and problems in number theory. This was one of the lines of Langlands' program, and mathematicians hoped that unsolved problems in number theory could be solved by studying the corresponding problems in geometry, which also remained unsolved. Such a program was known as the philosophy of concurrency. Those algebraic geometers who tried to solve problems in number theory were called "arithmetic algebraic geometers". In 1983, they heralded their first significant victory when Gerd Faltings of the Princeton Institute for Advanced Study made significant contributions to the understanding of Fermat's Theorem. Recall that, according to Fermat, the equation
at n greater than 2 has no solutions in integers. Faltings thought he had made progress in proving Fermat's Last Theorem by studying the geometric surfaces associated with different values n. Surfaces associated with Fermat's equations for various values n, differ from each other, but have one common property - they all have through holes, or, simply speaking, holes. These surfaces are four-dimensional, as are the graphs of modular forms. Two-dimensional sections of two surfaces are shown in fig. 23. The surfaces associated with Fermat's equation look similar. The greater the value n in the equation, the more holes in the corresponding surface.
Rice. 23. These two surfaces were obtained using the computer program Mathematica. Each of them represents the locus of points satisfying the equation x n + y n = z n(for the surface on the left n=3, for the surface on the right n=5). Variables x and y are considered to be complex.
Faltings was able to prove that, since such surfaces always have several holes, the associated Fermat equation could only have a finite set of solutions in integers. The number of solutions could be anything from zero, as Fermat suggested, to a million or a billion. Thus, Faltings did not prove Fermat's Last Theorem, but at least managed to reject the possibility that Fermat's equation could have infinitely many solutions.
Five years later, Miyaoka reported that he had gone one step further. He was then in his early twenties. Miyaoka formulated a conjecture about some inequality. It became clear that proving his geometric conjecture would mean proving that the number of solutions to Fermat's equation is not only finite, but zero. Miyaoka's approach was similar to Wiles' in that they both tried to prove Fermat's Last Theorem by relating it to a fundamental conjecture in another area of mathematics. For Miyaoka it was algebraic geometry, for Wiles the path to proof lay through elliptic curves and modular forms. Much to Wiles's chagrin, he was still struggling with the proof of the Taniyama-Shimura conjecture when Miyaoka claimed to have a complete proof of his own conjecture, and hence of Fermat's Last Theorem.
Two weeks after his speech in Bonn, Miyaoka published the five pages of calculations that formed the essence of his proof, and a thorough check began. Number theorists and algebraic geometries all over the world studied, line by line, published calculations. A few days later, mathematicians discovered one contradiction in the proof, which could not but cause concern. One part of Miyaoka's work led to a statement from number theory, from which, when translated into the language of algebraic geometry, a statement was obtained that contradicted the result obtained several years earlier. While this did not necessarily invalidate Miyaoka's entire proof, the discrepancy that was discovered did not fit into the philosophy of parallelism between number theory and geometry.
Two weeks later, Gerd Faltings, who paved the way for Miyaoke, announced that he had discovered the exact cause of the apparent violation of concurrency - a gap in reasoning. The Japanese mathematician was a geometer and was not absolutely strict in translating his ideas into the less familiar territory of number theory. An army of number theorists made desperate efforts to patch up the hole in Miyaoki's proof, but in vain. Two months after Miyaoka announced that he had a complete proof of Fermat's Last Theorem, the mathematical community came to the unanimous conclusion that Miyaoka's proof was doomed to failure.
As in the case of previous failed proofs, Miyaoka managed to obtain many interesting results. Parts of his proof deserve attention as very ingenious applications of geometry to number theory, and in later years other mathematicians used them to prove certain theorems, but no one succeeded in proving Fermat's Last Theorem in this way.
The hype about Fermat's Last Theorem soon died down, and the newspapers carried brief notes saying that the three-hundred-year-old puzzle still remained unsolved. On the wall of the New York subway station on Eighth Street appeared the following inscription, no doubt inspired by press publications about Fermat's Last Theorem: "The equation xn + yn = zn has no solutions. I have found a truly amazing proof of this fact, but I cannot write it down here because my train has come.
CHAPTER 10 CROCODILE FARM They drove along the scenic road in old John's car, sitting in the back seats. Behind the wheel was a black driver in a brightly colored shirt with an oddly cropped head. Bushes of black hair, hard as wire, rose on a shaved skull, logic
Race preparation. Alaska, Linda Pletner's Iditarod Farm is an annual dog sled race in Alaska. The length of the route is 1150 miles (1800 km). It is the longest dog sled race in the world. Start (ceremonial) - March 4, 2000 from Anchorage. Start
Goat Farm There is a lot of work in the village during the summer. When we visited the village of Khomutets, hay was being harvested and fragrant waves from freshly cut grass seemed to soak everything around. Grasses must be mowed in time so that they do not overripe, then everything valuable and nutritious will be preserved in them. This
Summer farm Straw, like lightning hand, into glass grass Another, having signed on the fence, lit the fire of the green glass of Water in the horse's trough. Into the blue dusk Wander, swaying, nine ducks along the rut of the spirit of parallel lines. Here is a chicken staring at nothing alone
Ruined farm The calm sun, like a flower of dark red, Went down to the earth, growing into the sunset, But the curtain of the night in idle power Twitched the world, which disturbed the look. Silence reigned on a farm without a roof, As if someone had torn off her hair, They fought over a cactus
Farm or backyard? On February 13, 1958, all the central Moscow and then regional newspapers published the decision of the Central Committee of the Communist Party of Ukraine "On an error in the purchase of cows from collective farmers in the Zaporozhye region." It was not even about the entire region, but about two of its districts: Primorsky
Fermat's problem In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. “At school, I loved solving problems, I took them home and came up with new ones from each problem. But the best problem I've ever come across, I found in a local
From the Pythagorean Theorem to Fermat's Last Theorem The Pythagorean theorem and the infinite number of Pythagorean triples were discussed in the book by E.T. Bell's "The Great Problem" - the same library book that caught the attention of Andrew Wiles. And although the Pythagoreans reached almost complete
Mathematics after the proof of Fermat's Last Theorem Oddly enough, Wiles himself had mixed feelings about his report: “The occasion for the speech was very well chosen, but the lecture itself aroused mixed feelings in me. Work on the proof
CHAPTER 63 Old McLennon's Farm About a month and a half after returning to New York on one of the "November evenings" the phone rang at the Lennons' apartment. Yoko picked up the phone. A Puerto Rican male voice asked Yoko Ono.
Pontryagin's theorem Simultaneously with the Conservatory, dad studied at Moscow State University, at the Mechanics and Mathematics. He successfully graduated from it and even hesitated for some time in choosing a profession. Musicology won, as a result of which he benefited from his mathematical mindset. One of my father's fellow students
Theorem The theorem on the right of a religious association to choose a priest needs to be proved. It reads like this: "An Orthodox community is being created... under the spiritual guidance of a priest chosen by the community and having received the blessing of the diocesan bishop."
I. Farm (“Here, from chicken manure…”) Here, from chicken manure One salvation is a broom. Love - which counts? - They took me to the chicken coop. Pecking at the grain, the hens cackle, the roosters march importantly. And without size and censorship Poems are composed in the mind. About a Provençal afternoon
Since few people know mathematical thinking, I will talk about the largest scientific discovery - the elementary proof of Fermat's Last Theorem - in the most understandable, school language.
The proof was found for a particular case (for a prime power n>2), to which (and the case n=4) all cases with composite n can be easily reduced.
So, we need to prove that the equation A^n=C^n-B^n has no solution in integers. (Here the ^ sign means degree.)
The proof is carried out in a number system with a simple base n. In this case, in each multiplication table, the last digits are not repeated. In the usual, decimal system, the situation is different. For example, when multiplying the number 2 by both 1 and 6, both products - 2 and 12 - end in the same numbers (2). And, for example, in the septenary system for the number 2, all the last digits are different: 0x2=...0, 1x2=...2, 2x2=...4, 3x2=...6, 4x2=...1, 5x2=...3, 6x2=...5, with a set of last digits 0, 2, 4, 6, 1, 3, 5.
Thanks to this property, for any number A that does not end in zero (and in Fermat's equality, the last digit of the numbers A, well or B, after dividing the equality by the common divisor of the numbers A, B, C is not equal to zero), you can choose a factor g such that the number Ag will have an arbitrarily long ending like 000...001. It is by such a number g that we multiply all the base numbers A, B, C in Fermat's equality. At the same time, we will make the single ending long enough, namely, two digits longer than the number (k) of zeros at the end of the number U=A+B-C.
The number U is not equal to zero - otherwise C \u003d A + B and A ^ n<(А+В)^n-B^n, т.е. равенство Ферма является неравенством.
That, in fact, is the whole preparation of Fermat's equality for a brief and final study. The only thing we still have to do: we rewrite the right side of Fermat's equality - C ^ n-B ^ n - using the school expansion formula: C ^ n-B ^ n \u003d (C-B) P, or aP. And since further we will operate (multiply and add) only with the digits of the (k + 2)-digit endings of the numbers A, B, C, then we can ignore their head parts and simply discard them (leaving only one fact in memory: the left side of Fermat's equality is a POWER).
The only other thing worth mentioning is the last digits of the numbers a and P. In Fermat's original equality, the number P ends in the number 1. This follows from the formula of Fermat's little theorem, which can be found in reference books. And after multiplying the Fermat equality by the number g ^ n, the number P is multiplied by the number g to the power of n-1, which, according to Fermat's little theorem, also ends in the number 1. So in the new Fermat equivalent equality, the number P ends in 1. And if A ends in 1, then A^n also ends in 1, and therefore the number a also ends in 1.
So, we have a starting situation: the last digits A", a", P" of the numbers A, a, P end in the number 1.
Well, then a sweet and fascinating operation begins, called in preference a "mill": introducing into consideration the subsequent digits a "", a """ and so on, the numbers a, we exclusively "easily" calculate that they are also equal to zero! I put "easy" in quotation marks, because humanity could not find the key to this "easy" for 350 years! And the key really turned out to be unexpectedly and dumbfoundingly primitive: the number P must be represented as P = q ^ (n-1) + Qn ^(k + 2) It is not worth paying attention to the second term in this sum - after all, in the further proof we discarded all the numbers after the (k + 2)th in the numbers (and this drastically simplifies the analysis)! So after discarding the head parts numbers, Fermat's equality takes the form: ...1=aq^(n-1), where a and q are not numbers, but only the endings of the numbers a and q! (I do not introduce new notation, as this makes reading difficult.)
The last philosophical question remains: why can the number P be represented as P=q^(n-1)+Qn^(k+2)? The answer is simple: because any integer P with 1 at the end can be represented in this form, and IDENTICALLY. (You can think of it in many other ways, but we don't need to.) Indeed, for P=1 the answer is obvious: P=1^(n-1). For P=hn+1, the number q=(n-h)n+1, which is easy to verify by solving the equation [(n-h)n+1]^(n-1)==hn+1 by two-valued endings. And so on (but we have no need for further calculations, since we only need the representation of numbers of the form P=1+Qn^t).
Uf-f-f-f! Well, philosophy is over, you can move on to calculations at the level of the second class, unless you just remember Newton's binomial formula once again.
So, let's introduce the number a"" (in the number a=a""n+1) and use it to calculate the number q"" (in the number q=q""n+1):
...01=(a""n+1)(q""n+1)^(n-1), or...01=(a""n+1)[(n-q"")n+ 1], whence q""=a"".
And now the right side of Fermat's equality can be rewritten as:
A^n=(a""n+1)^n+Dn^(k+2), where the value of the number D does not interest us.
And now we come to the decisive conclusion. The number a "" n + 1 is a two-digit ending of the number A and, THEREFORE, according to a simple lemma, it uniquely determines the THIRD digit of the degree A ^ n. And moreover, from the expansion of Newton's binomial
(a "" n + 1) ^ n, given that each term of the expansion (except the first, which the weather can no longer change!) is joined by a SIMPLE factor n (the base of the number!), It is clear that this third digit is equal to a "" . But by multiplying Fermat's equality by g ^ n, we turned the k + 1 digit before the last 1 in the number A into 0. And, therefore, a "" \u003d 0 !!!
Thus, we completed the cycle: by introducing a"", we found that q""=a"", and finally a""=0!
Well, it remains to be said that after carrying out completely similar calculations and the subsequent k digits, we obtain the final equality: (k + 2)-digit ending of the number a, or C-B, - just like the number A, is equal to 1. But then the (k+2)-th digit of C-A-B is equal to zero, while it is NOT equal to zero!!!
Here, in fact, is all the proof. To understand it, you do not need to have a higher education and, moreover, to be a professional mathematician. However, professionals keep quiet...
The readable text of the full proof is located here:
Reviews
Hello Victor. I liked your resume. "Don't let die before death" sounds great, of course. From the meeting in Prose with Fermat's theorem, to be honest, I was stunned! Does she belong here? There are scientific, popular science and teapot sites. Otherwise, thank you for your literary work.
Sincerely, Anya.
Dear Anya, despite the rather strict censorship, Prose allows you to write ABOUT EVERYTHING. With Fermat's theorem, the situation is as follows: large mathematical forums treat fermatists obliquely, with rudeness and, on the whole, treat them as best they can. However, in small Russian, English and French forums, I presented the last version of the proof. Nobody has put forward any counterarguments yet, and, I am sure, no one will put forward (the proof has been checked very carefully). On Saturday I will publish a philosophical note about the theorem.
There are almost no boors in prose, and if you don’t hang around with them, then pretty soon they come off.
Almost all of my works are presented in Prose, so I also placed the proof here.
See you later,
File FERMA-KDVar © N. M. Koziy, 2008
Certificate of Ukraine No. 27312
A BRIEF PROOF OF FERMAT'S GREAT THEOREM
Fermat's Last Theorem is formulated as follows: Diophantine equation (http://soluvel.okis.ru/evrika.html):
BUT n + V n = C n * /1/
where n- a positive integer greater than two has no solution in positive integers A , B , FROM .
PROOF
From the formulation of Fermat's Last Theorem it follows: if n is a positive integer greater than two, then, provided that two of the three numbers BUT , AT or FROM are positive integers, one of these numbers is not a positive integer.
We build the proof on the basis of the fundamental theorem of arithmetic, which is called the “theorem on the uniqueness of factorization” or “theorem on the uniqueness of the factorization of integer composite numbers into prime factors”. Odd and even exponents possible n . Let's consider both cases.
1. Case One: Exponent n - odd number.
In this case, the expression /1/ is converted according to known formulas as follows:
BUT n + AT n = FROM n /2/
We believe that A and B are positive integers.
Numbers BUT , AT and FROM must be relatively prime numbers.
From equation /2/ it follows that for given values of numbers A and B factor ( A + B ) n , FROM.
Let's say the number FROM - a positive integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition :
FROM n = A n + B n =(A+B) n ∙ D n , / 3/
where is the multiplier D n D
From equation /3/ it follows:
Equation /3/ also implies that the number [ C n = A n + B n ] provided that the number FROM ( A + B ) n. However, it is known that:
A n + B n < ( A + B ) n /5/
Consequently:
is a fractional number less than one. /6/
Fractional number.
n
For odd exponents n >2 number:
< 1- дробное число, не являющееся рациональной дробью.
From the analysis of the equation /2/ it follows that with an odd exponent n number:
FROM n = BUT n + AT n = (A+B)
consists of two definite algebraic factors, and for any value of the exponent n the algebraic factor remains unchanged ( A + B ).
Thus, Fermat's Last Theorem has no solution in positive integers for an odd exponent n >2.
2. Case Two: Exponent n - even number .
The essence of Fermat's last theorem will not change if the equation /1/ is rewritten as follows:
A n = C n - B n /7/
In this case, the equation /7/ is transformed as follows:
A n = C n - B n = ( FROM +B)∙(C n-1 + C n-2 B+ C n-3 ∙ B 2 +…+ C ∙ B n -2 + B n -1 ). /8/
We accept that FROM and AT- whole numbers.
From equation /8/ it follows that for given values of numbers B and C factor (C+ B ) has the same value for any value of the exponent n , hence it is a divisor of a number A .
Let's say the number BUT is an integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition :
BUT n = C n - B n =(C+ B ) n ∙ D n , / 9/
where is the multiplier D n must be an integer and therefore a number D must also be an integer.
From equation /9/ it follows:
/10/
Equation /9/ also implies that the number [ BUT n = FROM n - B n ] provided that the number BUT- an integer, must be divisible by a number (C+ B ) n. However, it is known that:
FROM n - B n < (С+ B ) n /11/
Consequently:
is a fractional number less than one. /12/
Fractional number.
It follows that for an odd value of the exponent n equation /1/ of Fermat's last theorem has no solution in positive integers.
With even exponents n >2 number:
< 1- дробное число, не являющееся рациональной дробью.
Thus, Fermat's Last Theorem has no solution in positive integers and for an even exponent n >2.
The general conclusion follows from the above: the equation /1/ of Fermat's last theorem has no solution in positive integers A, B and FROM provided that the exponent n>2.
ADDITIONAL REASONS
In the case when the exponent n – even number, algebraic expression ( C n - B n ) decomposed into algebraic factors:
C 2 - B 2 \u003d(C-B) ∙ (C+B); /13/
C 4 – B 4 = ( C-B) ∙ (C+B) (C 2 + B 2);/14/
C 6 - B 6 =(C-B) ∙ (C + B) (C 2 -CB + B 2) ∙ (C 2 + CB + B 2) ; /15/
C 8 - B 8= (C-B) ∙ (C+B) ∙ (C 2 + B 2) ∙ (C 4 + B 4)./16/
Let's give examples in numbers.
EXAMPLE 1: B=11; C=35.
C 2 – B 2 = (2 2 ∙ 3) ∙ (2 23) = 2 4 3 23;
C 4 – B 4 = (2 2 ∙ 3) ∙ (2 23) (2 673) = 2 4 3 23 673;
C 6 – B 6 = (2 2 ∙ 3) ∙ (2 23) (31 2) (3 577) =2 ∙ 3 ∙ 23 ∙ 31 2 ∙ 577;
C 8 – B 8 = (2 2 ∙ 3) ∙ (2 23) (2 673) ∙ (2 75633) = 2 5 ∙ 3 ∙ 23 ∙673 ∙ 75633 .
EXAMPLE 2: B=16; C=25.
C 2 – B 2 = (3 2) ∙ (41) = 3 2 ∙ 41;
C 4 – B 4 = (3 2) ∙ (41) (881) =3 2 ∙ 41 881;
C 6 – B 6 = (3 2) ∙ (41) ∙ (2 2 ∙ 3) ∙ (13 37) (3 ∙ 7 61) = 3 3 7 ∙ 13 37 ∙ 41 ∙ 61;
C 8 – B 8 = (3 2) ∙ (41) ∙ (881) ∙ (17 26833) = 3 2 ∙ 41 ∙ 881 ∙ 17 26833.
From the analysis of equations /13/, /14/, /15/ and /16/ and their corresponding numerical examples, it follows:
For a given exponent n , if it's an even number, a number BUT n = C n - B n decomposes into a well-defined number of well-defined algebraic factors;
For any degree n , if it is an even number, in algebraic expression ( C n - B n ) there are always multipliers ( C - B ) and ( C + B ) ;
Each algebraic factor corresponds to a well-defined numerical factor;
For given values of numbers AT and FROM numeric factors can be prime numbers or composite numeric factors;
Each composite numerical factor is a product of prime numbers, which are partially or completely absent from other composite numerical factors;
The value of prime numbers in the composition of composite numerical factors increases with the increase in these factors;
The composition of the largest composite numerical factor corresponding to the largest algebraic factor includes the largest prime number to a power less than the exponent n(most often in the first degree).
CONCLUSIONS: additional justifications support the conclusion that Fermat's Last Theorem has no solution in positive integers.
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