Алексей Виноградов – Fermat's Last Theorem (Conditions and decisions) (страница 1)
Алексей Виноградов
Fermat's Last Theorem (Conditions and decisions)
1. Pierre de Fermat
Pierre Fermat
The French lawyer and mathematician Pierre de Fermat or Petri de Fermat (1601–1665), with his work had a great influence on the further development of mathematics.
Pierre Fermat is one of the founders of number theory. He took this important step in his works on the largest and smallest quantities, which opened a series of studies by Fermat, which is one of the largest links in the history of the development of not only higher analysis in general, but also the analysis of infinitesimals in particular.
Fermat made the following statement: if a number a is not divisible by a prime number p, then there is an exponent k such that a-1 is divisible by p, and k is a divisor of p-1. This statement is called Fermat's little theorem. It is fundamental in all elementary number theory.
Pierre Fermat, according to the rules accepted in mathematics of the 19th-21st centuries, found tangents to algebraic curves.
In mathematical analysis, Fermat's lemma or the necessary criterion for an extremum is used: at extremum points, the derivative of a function is equal to zero.
Fermat developed a method for systematically finding all divisors of a number and formulated a theorem on the possibility of representing an arbitrary number by a sum of no more than four squares.
In the field of the infinitesimal method, he systematically studied the process of differentiation, gave a general law for differentiation of powers, and applied this law to the differentiation of fractional powers. In the preparation of modern methods of differential calculus, his creation of the rule for finding extrema was of great importance.
P. Fermat formulated the general law of differentiation of fractional powers and extended the formula for integrating powers to the cases of fractional and negative exponents.
Pierre Fermat developed the foundations of probability theory. In Fermat's works, both basic processes of the infinitesimal method were systematically developed, but he ignored the connection between the operations of differentiation and integration. This connection was made later.
Fermat was the first to come up with the idea of coordinates and created analytical geometry, introducing an infinitesimal quantity. He solved the problem of squaring any curve, and on this basis solved a number of problems on finding the centers of gravity. In the work “Introduction to the Theory of Plane and Spatial Places,” he was the first to classify curves depending on the order of their equation, establishing that a first-order equation defines a straight line, and a second-order equation defines a conic section. Developing these ideas, he applied analytical geometry to space.
In the field of physics, Fermat is associated with the establishment of the basic principle of geometric optics, by virtue of which light in an inhomogeneous medium chooses the path that takes the least time (Fermat believed that the speed of light is infinite and formulated the principle more vaguely). With this thesis begins the history of the main law of physics - the principle of least action.
The first collected works of P. Fermat, “Various Works,” was published in 1679.
2. The problem of dividing a square into the sum of two squares
P. Fermat often wrote down not proofs, but only brief instructions about the method he used.
Page from the book “Arithmetic” by Diophantus 1670.From the author's archive
In one of Diophantus's books, the problem of dividing a square into the sum of two squares was considered, and he meant the squares of positive numbers. In the margins of the book, Fermat wrote:
«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».
«But the cube into two cubes, or a square-square into two square-squares, and in general it is right to divide any power infinitely beyond the square into two of the same name; Of course I discovered a wonderful demonstration of this matter. The smallness of the margin would not take him».
There was no general proof in Fermat's papers. He published a proof only for the case n = 4. A proof for n = 3 was given in 1768, for n = 5 in 1823, for n = 7 in 1837, for n = 67 in 1851, for powers up to 2521 in 1955, for degrees up to 4002 in 1966, for degrees up to 100000 in 1977.
And finally, at the end of the 20th century, through enormous efforts and calculations, the theorem was allegedly proven. Andrew John Wiles discovered a technical method that, with the help of Richard Taylor, allowed him to complete the proof in 1994. Wiles's proof, containing 129 pages, was published in the «Annals of Mathematics» in 1995.
Without touching on the fallacy of this solution, it can be noted that it cannot be the one sought by Pierre Fermat in the 17th century. The margins of the book were not wide enough for his entry, but still, it could fit on wider margins or a separate sheet, and not on 129 pages. The width of the book margin is 5 cm, the longest length is 63 cm (edge). Thus, the proof occupied 315 - 450 sq. cm of handwritten font, less than an A 4 page (623.7 sq. cm), while some of the characters (integral, etc.) were replaced by Latin words.
Maximum field width of the book of Diophantus
3. Wiles–Taylor proof
At the end of the 20th century, through enormous efforts and calculations, Fermat's theorem was allegedly proven.
Andrew John Wiles discovered a technical method that, with the help of Richard Taylor, allowed him to complete the proof in 1994. Wiles's proof, containing 129 pages, was published in the «Annals of Mathematics» in 1995.
Meanwhile, it was not Fermat’s Last Theorem that was proven, but part of the Taniyama-Shimura-Weil theorem (the statement states that every elliptic curve over the field of algebraic numbers is automorphic), a frequent case of which some mathematicians considered Fermat’s Last Theorem. In the process of reasoning, Fermat's original statement was reformulated in terms of comparing a Diophantine equation of the p – degree with elliptic curves of the 3rd order. (That is, no one was going to prove the original theorem). This comparison forced the authors of the proof to announce that their method and reasoning lead to a final solution to Fermat's problem.
Fermat's Last Theorem.From the author's archive
In 1999, Christophe Broglie, Brian Conrad, Fred Diamond, and Richard Taylor proved unstable cases of the Taniyama-Shimura-Weil theorem. Although they believe that they have proven the Taniyama-Shimura-Weil theorem, the scope of Wiles' method is limited. It only works for elliptic curves over rational numbers, while the Taniyama-Shimura-Weil conjecture covers elliptic curves over any algebraic number field. Therefore, it is assumed that there is a more general and more elegant proof of the modularity of elliptic curves.
Wiles considered the most general case of the Taniyama-Shimura-Weil theorem and proved this theorem for all semistable elliptic curves over the field of rational numbers in 1995, which he and his colleagues consider sufficient to prove Fermat's Last Theorem. In March 1996, Wiles and Robert Langlands received recognition from one of the institutions as the authors of the proof. Although neither of them completely proved the theorem, they were said to have made significant contributions that greatly facilitated further proof, which in 1996 the mathematical community believed they did not yet have.
Also, the proof of Fermat’s Last Theorem based on Taniyama’s theorem was attributed to 1993, 1995, 1998 or 2001. At the end of 2007, the scientific society “Göttingen Academy of Sciences” turned to some anonymous mathematical authorities and, having received from them good, awarded the prize for proving Fermat's Last Theorem to Wiles, and liquidated the bank account. The scientific community never learned the names of the mathematical authorities; out of modesty, they themselves did not tell us about themselves. As of 2024, no mathematician has ever held a published proof of Fermat's Last Theorem (based on Taniyama's theorem).
Wiles's work on the theorem was designed as a "proof without proof", that is, the proof is not direct and immediate. Wiles began working on Fermat's theorem after Ken Ribet showed the connection between semistable elliptic curves (a special case of the Taniyama-Shimura-Weil theorem) and Fermat's theorem. The main idea about such a connection belongs to the German mathematician Gerhard Frey. Fry suggested that the Taniyama-Shimura-Weil conjecture is a generalization of Fermat's Last Theorem, because any counterexample to Fermat's Last Theorem resulted in a non-modular elliptic curve.
Wiles in his proof proceeds from the fact that Fermat's theorem is a consequence of Taniyama's conjecture about modular elliptic formations. This conclusion was made on the basis of a limited number of points x, y, z from Fermat’s theorem, which allow the author to assert that these points characterize all combinations of x, y, z and n as being involved in modular elliptic curves.