Fermat's Last Theorem (Conditions and decisions)

Алексей Виноградов

Fermat's Last Theorem (Conditions and decisions)

Illustrations taken from the first edition of the book Vinogradov A. G. (in Russian) Великая Теорема Ферма (Условия и решения) – Fermat's Last Theorem (Conditions and decisions).

Pierre de 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.

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

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

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

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.

At the same time, it becomes unclear: either the validity of Taniyama’s conjecture is being proven using Fermat’s unproven theorem, or Fermat’s theorem is being proven using the unproven Taniyama-Shimura-Weil conjecture. The absence of a relationship between exponents of powers n > 2 and powers n = 1 and 2 raises doubts; the distribution of the conditions of Fermat’s theorem over the XOY plane and, in particular, to negative integers is not shown.

The conclusion about the truth of Fermat's equation is made on incorrect postulates.

First, it is assumed that there is some solution to Fermat's equation in positive integers.

Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (a plane curve of degree 3) under the assumption that the elliptic curves thus obtained exist (the second unconfirmed assumption).

Thirdly, since other methods prove that the particular curve constructed is not modular, it means that it does not exist. This leads to the conclusion: there is no integer solution to Fermat’s equation and, therefore, the theorem is true.

There is a discrepancy between Fermat's equations and algebraic (elliptic) curves of the 3rd order. Frey proposed that the supposed elliptic curve representing the hypothetical integer solution of Fermat's equation could not be modular.

Wiles decided that every semistable elliptic curve defined over the field of rational numbers is modular. This led to the conclusion about the impossibility of integer solutions of Fermat’s equation and, consequently, about the validity of Fermat’s statement. He refers to the work of I. Elleguarche, in which he found a way to associate a hypothetical third-order curve with Fermat's equation. The author conducts his reasoning in terms of projective geometry. The curve was a priori perceived as elliptical. However, the segments of the Euclidean line, when adding points on it, are taken on a non-linear scale, and the curve is not actually elliptic.

An appeal in this case to derive the Fermat equation seems illegitimate. Despite the fact that it satisfies some criteria for the class of elliptic curves, it does not satisfy the main criterion of being an equation of the 3rd degree in a linear coordinate system.

To date, there is not a single scientific objection to the recognition of the fallacy of Wiles's proof.

Fermat's Last Theorem and Soviet Science

It is believed that for some reason they did not try to solve Fermat’s Last Theorem in the USSR. This is not true. At one time, it became clear that in order to review a solution, a written review from a competent reviewer was required, and the review fell within the competence of the Order of Lenin Mathematical Institute. V.A. Steklov of the USSR Academy of Sciences (MIAN). And in the presence of such support (Doctor of Mathematical Sciences), it turned out that this problem does not relate to computational mathematics and in general the Steklov Mathematical Institute “has never advised anyone and does not advise anyone to deal with the Fermat problem,” and its “Scientific Council decided not to consider any more materials devoted to this problem ". We are not looking at the problem, and the problem is gone!

True, in Soviet times they at least answered, but in the new Russia they no longer do this. It is not surprising that in the end Russian science has reached the modern level, for which it is ready to blame anyone but itself.

However, the problem of proving Fermat's Last Theorem was considered not only by academicians Postnikov and Fomenko, but also by less titled people. Let us note that, in the opinion of «Izvestia ANSSSR», most, not all, of the proofs of Fermat’s Last Theorem that came to them turned out to be incorrect. And a minority, clearly more than one, of the decisions were correct. However, Soviet science, being deeply secondary to the West, did not allow itself to get ahead of it in anything.

Known published proofs do not match the dimensions of Pierre Fermat's proof, and in most cases they use mathematical apparatus developed after 1680. We will not argue how correct these constructions turned out to be (we will present them further).

Fermat's Last Theorem – proof options

Some decisions were published in the republics of the USSR. Although the mathematical journals of the Russian Academy of Sciences, «Algebra and Analysis», for example, do not consider elementary proofs of Fermat’s theorem, other departments of the Academy of Sciences were not prohibited from publishing. As far as is known, other decisions were published in the USSR.

It is impossible to determine how correct they are, because inaccuracies may have been made during publication.

First proof 1993-2012

Bobrov A.V. in 1993-2012 wrote a series of proofs of Fermat’s theorem. Option No. 1 was published in the journal of the Russian Academy of Sciences “Questions in the history of natural science and technology” No. 3, 1993.

Fermat's Last Theorem – two short proofs

Fermat's last theorem, sometimes called the Great Theorem, is formulated as follows: in the equality an + bn = cn, the numbers an, bn, cn and n cannot simultaneously be positive integers if n > 2.

Let's assume such numbers exist. Then the following conditions must be met:

The equality is valid for relatively prime numbers a, b and c that do not have common integer factors other than 1, i.e. two numbers are always odd.

There are numbers d1= c – b and d2 = c – a, or d1 – d2 = a – b, that is, for arbitrarily chosen natural numbers a > b there is an infinite set of rational, real or complex numbers d1 and d2 satisfying the above equality, if Arithmetic operations are feasible in this set. For integers c, the numbers d1 and d2 will also be integers.

Option 1

The equality cn – bn – an = 0 (1) by sequential division by numbers d1 and d2 is always transformed into two polynomials (equations) (n-1)th degree with respect to c:

(cn – bn)/ d1 – an/ d1 = cn-1+ cn-2 x b+…+c x bn-2 + bn-1 – (an/ d1) = 0 (2)

(cn – an)/ d2 – bn/ d2 = cn-1+ cn-2 x a+…+c x an-2 + an-1 – (bn/ d2) = 0 (3)

Equalities (2) and (3) were obtained by identical transformations of equality (1), i.e. must be executed for the same values of positive integers a, b and c. By definition, a necessary and sufficient condition for the identity of two polynomials over a certain number field (in our case, over a set of integers) is the equality of the coefficients of terms containing the same arguments in the same powers, that is, the following must be true:

a = b, a2 = b2, … an-2 = bn-2, (an/ d1) – bn-1 = (bn / d2) – an-1 (4)

From (1) and (4) it follows that cn = 2an, c =an√2, that is, the number c, as the common arithmetic root of equations (1), (2) and (3), cannot be rational for integer a, b. d1 and d2.

From the equality of free members it follows:

(an – (bn-1 x c) + bn ) / (c-b) = (bn – (an-1 x c) + an ) / (c-a), or

(c x (cn-1 – bn-1) / (c-b) = (c x (cn-1 – an-1) / (c-a), or

cn-2 + сn-3 x b +…+ с x bn-3 +bn-2 = cn-2 + сn-3 x a +…+ с x an-3 +an-2 (5)

Subtracting the left side from the right side of equality (5), we obtain:

cn-3 (a – b)+ cn-4 x (a2 – b2 ) +… + c (an-3 – bn-3) + (an-2 – bn-2 ) = 0 (6)

or, if a ≠ b, reducing by a – b ≠ 0, we get:

cn-3 + cn-4 x (a + b ) +… + c (an-4 +… + bn-4) +(an-3 +… + bn-3 ) = 0 (7)

From equality (7) it follows that for a ≠ b the numbers a, b and c cannot be simultaneously positive.

The presented transformations allow us to draw the following conclusions:

– for polynomials (2) and (3) that are identical over the set of rational numbers for n > 2, the number c is the common arithmetic root of equations (1), (2) and (3), cannot be rational for positive integers a , b , d1 and d2;

– polynomials (2) and (3) for n > 2 and natural a and b are not identical over the set of rational numbers if the divisors d1 and d2 of equality (1) are irrational, which implies that the number c is irrational;

– the numbers a, b and c in equality (1) for n > 2 cannot be rational at the same time.

For n ≤ 2, the contradiction disappears, the coefficients of cn-1 are equal to 1, and the equality of free terms after substituting the values d1 and d2 becomes an identity:

(a2 – bc + b2 ) / c – b =(b2 – ac + a2 ) / c – a (8)

If the right and left sides of equality (5) are denoted respectively by q1 and q, where q1 and q are positive integers, then polynomials (2) and (3) are transformed into quadratic equations with respect to c:

q x2 – (b q – b n-1) x – (an + bn ) = 0

q1 x2 – (a q1 – a n-1) x – (an + bn ) = 0 (9)

where the unknown c is denoted in the generally accepted way by x, that is, x = c

The same conclusions follow from the conditions of equivalence or analysis of the reasons for the non-equivalence of these equations.

Option 2

Let in the equality cn – bn = an the numbers a, b and c be relatively prime, a ≠ 1 – be an odd number. For any positive numbers, we can perform the operation of finding the arithmetic value of the square root, that is, we can write:

(cn/2 + bn/2) x (cn/2 – bn/2) = a1n x a2n = an (1)

Where a1n = cn/2 + bn/2 , a2n = cn/2 – bn/2 are real positive factors of the number an.

From (1) it follows: cn = (an1 + an2)/2)2, bn = (an1 – an2)/2)2 (2)

In accordance with the properties of the exponential function, for real positive numbers a1n , a2n and integer a ≠ 1, there are unique values of the exponents – <∞ q < p <+∞,, satisfying the equalities:

a n1 = a p, a2n = a q (3) , где a = a1 a2, n = p + q.

From (3) it follows a1p a1q = a1pa2p, a2p a2q = a1qa2q , or after reduction by numbers

a1p ≠ 0 , a2q ≠ 0 получим: a1 q = a2 p (4)

From (1), (2) and (3) it follows:

cn – bn = (ap +aq)/2)2 – (ap -aq)/2)2 = ap+q =an (5)

or, taking into account equalities (3) and (4):

(a1 p * a1 q +a1 q * a 1q)/2)2 – (a1 p * a1 q – a1 q * a 1q)/2)2 = a1 p * a1 2q * a2 q (6)

Let's take the common factor a1q out of brackets:

a1 2q * (a1 p +a2 q )/2)2 – a1 2q * (a1 p – a2q )/2)2 = a1 p*a1 2q * a2 q (7)

From (5) and (7) it follows that the numbers cn , bn and an contain a common factor a1 2q = a22 p ≠ 0, which contradicts the condition of their mutual simplicity, if a1 2q = a22 p ≠ 1. From a1 2q = a22 p = 1 it follows q = 0 , a2 = 1, that is, n = p , a = a1 *1, and equalities (5) and (7) take the form: cn – bn = (an +1 )/2)2 – (an -1 )/2)2 = an (8)