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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Fermat's Last Theorem. History and essence of scientific discovery
Directory / The most important scientific discoveries One of Pierre de Fermat's obituaries said: "He was one of the most remarkable minds of our century, such a universal genius and so versatile that if all scientists did not pay tribute to his extraordinary merits, it would be difficult to believe all the things that need to be said about him. to say so as not to miss anything in our eulogy." Unfortunately, not much is known about the life of the great scientist. Pierre Farm (1601-1665) was born in the south of France in the small town of Beaumont-de-Lomagne, where his father, Dominique Fermat, was the "second consul", that is, assistant to the mayor. Dominique Fermat gave his son a very solid education. In the college of his native city, Pierre acquired a good knowledge of languages: Latin, Greek, Spanish, Italian. He subsequently wrote poetry in Latin, French and Spanish. Fermat was famous as a fine connoisseur of antiquity, he was consulted about difficult places in the editions of the Greek classics. However, Pierre directed all the strength of his genius to mathematical research. Yet mathematics did not become his profession. Scientists of his time did not have the opportunity to devote themselves entirely to their beloved science. The farm elects jurisprudence. A bachelor's degree was awarded to him in Orleans. Since 1630, Fermat moved to Toulouse, where he received a position as an adviser in Parliament (i.e., the court). About his legal activities, it is said in a "commendable word" that he performed it "with great conscientiousness and such skill that he was famous as one of the best lawyers of his time." During Fermat's lifetime, his mathematical work became known mainly through the extensive correspondence he had with other scientists. The collected works, which he repeatedly tried to write, were never created by him. Yes, this is not surprising given the hard work in court that he had to perform. None of his writings was published during his lifetime. However, he gave several treatises a completely finished look, and they became known in manuscript to most of his contemporary scholars. In addition to these treatises, his extensive and extremely interesting correspondence remained. In the XNUMXth century, when there were no special scientific journals, correspondence between scientists played a special role. It set tasks, reported on methods for solving them, and discussed acute scientific issues. Fermat's correspondents were the greatest scientists of his time: Descartes, Etienne Pascal and Blaise Pascal, de Beesi, Huygens, Torricelli, Vallis. Letters were sent either directly to the correspondent, or to Paris, to the Abbé Mersenne (a fellow student of Descartes in college); the latter multiplied them and sent them to those mathematicians who dealt with similar questions. One of the first mathematical works of Fermat was the restoration of two lost books of Apollonius "On Flat Places". Fermat's great service to science is usually seen in his introduction of an infinitesimal quantity into analytic geometry, just as it was done a little earlier. Kepler regarding the geometry of the ancients. He took this important step in his 1629 work on the largest and smallest quantities, works that opened one of Fermat's most important series of studies, which are one of the largest links in the history of the development of not only higher analysis in general, but also analysis of infinitesimals in particular. At the end of the twenties, Fermat discovered methods for finding extrema and tangents, which, from a modern point of view, come down to finding a derivative. In 1636, the completed presentation of the method was transferred to Mersenne, and everyone could get acquainted with him. Before Fermat, the Italian scientist Cavalieri developed systematic methods for calculating areas. But already in 1642, Fermat discovered a method for calculating areas bounded by any "parabolas" and any "hyperbolas" He showed that the area of an unlimited figure can be finite. Fermat was one of the first to tackle the problem of straightening curves, that is, calculating the length of their arcs. He managed to reduce this problem to the calculation of some areas. Thus, Fermat's concept of "area" acquired a very abstract character. Problems of straightening curves were reduced to the determination of areas, he reduced the calculation of complex areas with the help of substitutions to the calculation of simpler areas. There was only a step left to pass from the area to the even more abstract concept of "integral". Fermat has many other accomplishments. He first came to the idea of coordinates and created analytical geometry. He also dealt with the problems of probability theory. But Fermat was not limited to mathematics alone, he also studied physics, where he owns the discovery of the law of propagation of light in media. Despite the lack of evidence (of which only one has survived), it is difficult to overestimate the importance of Fermat's work in the field of number theory. He alone managed to single out from the chaos of problems and particular questions that immediately arise before the researcher when studying the properties of integers, the main problems that became central to the entire classical theory of numbers. He also owns the discovery of a powerful general method for proving number-theoretic propositions - the so-called method of indefinite or infinite descent, which will be discussed below. Therefore, Fermat can rightfully be considered the founder of number theory. In a letter to de Bessy dated October 18, 1640, Fermat made the following statement: if the number а not divisible by a prime number р, then there is such an indicator к that а - divided by р, where k is a divisor р-one. This statement is called Fermat's little theorem. It is fundamental in all elementary number theory. Euler gave this theorem several different proofs. In the second book of his Arithmetic, Diophantus set the task of representing a given square as the sum of two rational squares. In the margins, against this task, Fermat wrote: “On the contrary, it is impossible to decompose neither a cube into two cubes, nor a biquadrate into two biquadrates, and in general to any power greater than a square, into two powers with the same exponent. I discovered a truly wonderful proof for this, but these fields are too narrow for him.” This is the famous Great Theorem. This theorem had an amazing fate. In the last century, her research has led to the construction of the most subtle and beautiful theories relating to the arithmetic of algebraic numbers. It can be said without exaggeration that it played no less a role in the development of number theory than the problem of solving equations in radicals. The only difference is that the latter has already been solved by Galois, and the Great Theorem still encourages mathematicians to research. On the other hand, the simplicity of the formulation of this theorem and the cryptic words about its "miraculous proof" led to the widespread popularity of the theorem among non-mathematicians and the formation of a whole corporation of "fermatists" who, in the words of Davenport, "have courage far beyond their mathematical abilities." Therefore, the Great Theorem is in first place in terms of the number of incorrect proofs given to it. Fermat himself left a proof of the Great Theorem for the fourth powers. Here he applied a new method. Fermat writes that "since the usual methods found in books were insufficient to prove such difficult propositions, I finally found a very special way to achieve them. I called this method of proof infinite or indefinite descent." It was by this method that many propositions of number theory were proved, and, in particular, with its help, Euler proved the Great Theorem for n=4 (in a way somewhat different from Fermat's method), and 20 years later for n=3. Fermat described this method in his letter to Karkavy (August 1659) as follows: "If there were some right-angled triangle in integers, which would have an area equal to the square, then there would be another triangle, smaller than this, which would have the same property. If there were a second, smaller than the first, which would have the same property, then there would exist, by virtue of this reasoning, a third less than the second, which would have the same property, and, finally, a fourth, fifth, descending to infinity.But if a number is given, then there is no I mean whole numbers.) Whence it is concluded that there is no right-angled triangle with a square area. Fermat goes on to say that, after much deliberation, he was able to apply his method to the proof of other affirmative propositions. “But to apply the method to the proof of other propositions,” writes I.G. Bashmakova, “for example, to prove that each number can be represented by a sum of no more than four squares, the application of “new principles” is required, on which Fermat does not dwell in more detail. a listing of all the theorems that Fermat proved using the descent method, including the great theorem for the case n = 3. At the end of the letter, Fermat expresses the hope that this method will be useful to subsequent mathematicians and show them that "the ancients did not know everything" "Unfortunately, this letter was published only in 1879. However, Euler restored Fermat's method from separate remarks and successfully applied it to problems of indefinite analysis. In particular, he also owns the proof of the grand theorem for n = 3. Recall that the first an attempt to prove the indecomposability of the cube of a natural number into the sum of two cubes was made around the year 1000 in the Arab East. The descent method again began to play a leading role in research on Diophantine analysis by A. Poincaré and A. Weyl. At present, to apply this method, the concept of height is introduced, that is, such a natural number, which in a certain way is put in correspondence with each rational solution. Moreover, if it is possible to prove that for each rational solution of height A there is another solution of height less than A, then this will imply the unsolvability of the problem in rational numbers. All subsequent algebraic number theory up to the papers Gaussian developed, starting from Fermat's problems. In the 5500th century, research related to Fermat's Last Theorem and the laws of reciprocity required an expansion of the field of arithmetic. Kummer, while working on Fermat's Last Theorem, built arithmetic for algebraic integers of a certain kind. This allowed him to prove the Great Theorem for a certain class of prime exponents n. At present, the validity of the Great Theorem has been verified for all exponents n less than XNUMX. We also note that the Great Theorem is connected not only with algebraic number theory, but also with algebraic geometry, which is now being intensively developed. But the Great Theorem in general form has not yet been proven. Therefore, we have the right to expect here the emergence of new ideas and methods. Author: Samin D.K.
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