The proof

of the Great theorem of Fermat

in a general view and its graphic representation

 1. The first variant of the proof

 2. The second variant of the proof

 3. Expansion of the Great theorem of Fermat

 © Lemyakin Boris Aleksandrovich

 email: lemyakin@yandex.ru

Debatable page (RUS)

Russian

 In 1630 the French mathematician - the fan, the lawyer by a trade Pierre Fermat (1601-1665) has written down on fields of Arithmetics of Diofant: "it is impossible to spread out neither a cube on two cubes, nor a biquadrate on two biquadrates and in general any degree, it is more than square, on two degrees with the same parameter" and has added: " I have opened to this really the wonderful proof, but these fields for its are too small " (the Mathematical encyclopedia. M. СЭ.1985.т.5, стр. 605-608). After his death in P.Fermat's papers have found the proof only for a degree equal 4.
 It is known, that P.Fermat in the mathematical researches counted valuable only a conclusion, result of search. At his life it has not been published any work. However scientists of that time in many respects are obliged P. Fermat which results of researches became known due to correspondence and personal meetings. In correspondence it quite often formulated only the theorem, believing excessive reasonings under its proof. Therefore, when his conclusions found acknowledgement, scientists should find considerable work proofs. Some of them have been proved already after death P. Fermat. And the Great theorem of Ferma as she is called by descendants, in that kind as she has been formulated by Pierre Fermat, is not proved till now.
Now the Great theorem of Fermat is formulated so: " For any natural number n>2 equation xn + yn = zn  has no decisions in the whole nonzero numbers x, y, z " (the Mathematical encyclopedia. M. СЭ.1985.т.5, стр. 605-608). Comparison of this formulation with the formulation of Fermat testifies that descendants not only have stated her more correctly, but also have essentially simplified. In this kind of the proof to it, under some data, are found, but "wonderful" them do not name. The proof recognized as the mathematical world American Andrew Wiles's mathematics by means of his colleague Taylor, using theory of Ivasave (section of the theory of numbers), hypothesis of Tanijama (analogue of the Great theorem which establishes communication between two-dimensional and four-dimensional forms), transformations Frey (leading the equation of the Great theorem to to the equation of an elliptic curve which appears in hypothesis of Tanijama), it is necessary not less than hundred pages. For certain Fermat meant other proof which corresponded to the mathematical device of that time.
It is known, that during this period when P. Fermat has formulated the theorem, it was engaged in researches of conic sections and infinitesimal sizes, has given the general law of differentiation and has applied it the law to differentiation of fractional degrees. Therefore it is quite probable, that it has deduced the equation, investigating characteristics of the truncated cone. Having armed with these data, I have decided to investigate only those ways with which could go Pierre Fermat.
 

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The first variant of the proof

In the rectangular triangle having the parties x, y, z₁ (fig. 1), is carried out equality

               z₁² = x² + y²                                                          (1)

fig.1

        At an exponent n>2 

               z₁ⁿ = (x² + y²)^n/2 > xⁿ + yⁿ                                         (2)

     It is obvious, that in the formula zⁿ = xⁿ + yⁿ                          (3)

                z > y ≥ x or z > x ≥ y                                     

     Thus, it is possible to ascertain, that to equality

                zⁿ = xⁿ + yⁿ at n>2

     there corresponds a figure, shall name her "not connected a rectangular triangle", with the parties x, y, z, at which party

                z < z₁                                                                       (4)

     The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle

                  z² < x² + y²                                                            (5)

     To conditions (3) and (5) satisfies also the acute-angled triangle having the parties x, y, z and opposite to the party z a corner z, and and

                   π/3 < z < π/2                                                           (6)

     This triangle can be received by connection of the parties of the opened rectangular triangle.

     The decision of the received acute-angled triangle concerning the party z

                    z² = x² + y² – 2xycos z                                           (7)

               Thence follows 

                zⁿ = (x² + y² – 2xycos z)^n/2                                         (8)

         In result it is possible to write down

                zⁿ = xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2                           (9)

     The great theorem the Farm in interpretation of a quantitative parity of number of individual objects has identity in geometrical interpretation of a parity of length of the parties of a triangle.

     We shall consider, what values trigonometrical function of a corner z can accept.

               cos z = (x² + y² - z²)/2xy = β, здесь 0 < β < 0,5          (10) 

     Value of a corner z is the attitude of length of the arch concluded between its parties which is described by any radius from a vertex of angle, to radius. The length of an arch is expressed by irrational number. Rational numbers it is possible to set only an interval of values in which there is its length. Therefore and value of a corner z can be set with any accuracy rational numbers only an interval of values in which there is its size. From here follows, that the corner z cannot be set a rational number. Values of trigonometrical function of this corner can be calculated for the set limits of an interval of values of a corner. But value of trigonometrical function of a corner cannot be expressed a rational number. Therefore values of length of a line of a sine and cosine also can be expressed rational numbers only values of limits of an interval in which these lengths are.

     In calculations sometimes it is used also градусное measurement of corners. Thus the equal-sign between irrational and a rational number, for example, is put       z = π/2 = 90⁰. This equality in the theoretical plan incorrectly. Also in the theoretical plan it is incorrect to express results of calculations of length of lines of sine and косинусов rational numbers. Rational numbers it is possible to express values only discretely changing sizes. Values of the functions which are not having breaks, in all cases are expressed by irrational numbers.

     If to assume, that cos z can accept rational values, and an equal-sign in (9) it is lawful, the Great theorem the Farm is denied. But then it is necessary to recognize incorrect the proof executed by Andrew Wiles's. For check of this conclusion it is possible to find set of decisions of acute-angled triangles in rational numbers concerning the party z. At substitution of the received values in (3) equality should take place, but it contradicts results of numerous experiments, computer calculations up to extreme great values x, y, z and n.

     In the modern mathematics it is accepted at transition from an acute-angled triangle to a rectangular triangle to equate value 2xycosz to zero. These actions are admissible at the decision of practical problems, but not admissible in the theoretical analysis as contradict (10).

     Strictly speaking, rectangular triangles do not exist at all. But, considering, that the main purpose of mathematics consists in service of applied sciences, it is conditionally possible to admit existence of the rectangular triangles set with any necessary accuracy. This assumption is used as a method of the decision of the equations, based on casual concurrence of a kind of the equations of the different sort describing parities of the areas and a parity of the parties of a triangle.

     In the theoretical plan it is possible to apply Pifagor theorem for definition of a parity of the areas, there where in the equation values of commensurable pieces are used. In triangles of length of the parties are incommensurable. At least, one of the parties has irrational value. It is proved also with the Great theorem of Fermat. It is the decision of the triangle which has been written down in the algebraic form.

     Considering told, there is no value zⁿ which would satisfy to equality (9).

               zⁿ ≠ xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2    at  n>2             (11)

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The second variant of the proof

In the rectangular triangle having the parties x, y, z₁ (fig. 1), is carried out equality

               z₁² = x² + y²                                                          (1)

fig.1

        At an exponent n>2 

               z₁ⁿ = (x² + y²)^n/2 > xⁿ + yⁿ                                         (2)

     It is obvious, that in the formula zⁿ = xⁿ + yⁿ                          (3)

                z > y ≥ x or z > x ≥ y                                     

     Thus, it is possible to ascertain, that to equality

                zⁿ = xⁿ + yⁿ at n>2

     there corresponds a figure, shall name her "not connected a rectangular triangle", with the parties x, y, z, at which party

                z < z₁                                                                       (4)

     The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle

                  z² < x² + y²                                                            (5)

     To conditions (3) and (5) satisfies also the acute-angled triangle having the parties x, y, z and opposite to the party z a corner z, and and

                   π/3 < z < π/2                                                           (6)

     This triangle can be received by connection of the parties of the opened rectangular triangle.

     The decision of the received acute-angled triangle concerning the party z

                    z² = x² + y² – 2xycos z                                           (7)

               Thence follows 

                zⁿ = (x² + y² – 2xycos z)^n/2                                         (8)

         In result it is possible to write down

                zⁿ = xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2                           (9)

     The Great theorem of Fermat in interpretation of a quantitative parity of number of individual objects has identity in geometrical interpretation of a parity of length of the parties of a triangle.

Triangle, according to (5), it is possible to transform to a rectangular triangle with the parties zⁿ, xⁿ and yⁿ multiplication of length of each of the parties to factors z^n-1, x^n-1 and y^n-1 accordingly. Its decision will look like

               z²ⁿ = x²ⁿ + y²ⁿ                                                           (10)

     If the parties of this triangle to reduce up to sizes z^n/2, x^n/2 and y^n/2  by return analogy with (2) we shall receive the opened rectangular triangle in which decision

                    zⁿ < xⁿ + yⁿ                                                           (11)

     But this value zⁿ us is already received by algebraic transformation (9).

     The Great theorem of Fermat is denied, if conditions (9) and (11) are simultaneously satisfied, that is

               zⁿ < xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2                              (12)

     It is obvious, that (12) contradicts (9). The condition of a refutation of the Great theorem of Fermat is not executed, hence, this theorem is true.

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Expansion of the Great theorem of Fermat

     Having assumed as a basis the first variant of the proof, we shall consider expression (11)     

     zⁿ ≠ xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2    in the field of values n≤2.

     At n=2 the acute-angled triangle will be transformed to a rectangular triangle.

     In the field of values 1<n<2 we have an obtusangular triangle, at which corner, opposite to the party z, z>π/2. In this area the Great theorem of Fermat will look like

                               zⁿ ≠ xⁿ + yⁿ      при 1<n<2                               (1)

fig.1

        At an exponent n>2 

               z₁ⁿ = (x² + y²)^n/2 > xⁿ + yⁿ                                         (2)

     It is obvious, that in the formula zⁿ = xⁿ + yⁿ                          (3)

                z > y ≥ x or z > x ≥ y                                     

     Thus, it is possible to ascertain, that to equality

                zⁿ = xⁿ + yⁿ at n>2

     there corresponds a figure, shall name her "not connected a rectangular triangle", with the parties x, y, z, at which party

                z < z₁                                                                       (4)

     The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle

                  z² < x² + y²                                                            (5)

     To conditions (3) and (5) satisfies also the acute-angled triangle having the parties x, y, z and opposite to the party z a corner z, and and

                   π/3 < z < π/2                                                           (6)

     This triangle can be received by connection of the parties of the opened rectangular triangle.

     The decision of the received acute-angled triangle concerning the party z

                    z² = x² + y² – 2xycos z                                           (7)

               Thence follows 

                zⁿ = (x² + y² – 2xycos z)^n/2                                         (8)

         In result it is possible to write down

                zⁿ = xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2                           (9)

     The Great theorem of Fermat in interpretation of a quantitative parity of number of individual objects has identity in geometrical interpretation of a parity of length of the parties of a triangle.

     The proof of an inequality

               zⁿ ≠ xⁿ + yⁿ = (x² + y² – 2xycos z)^n/2  at  1<n<2         (10)

does not differ from the first and second variant of the proof of the Great theorem of Fermat.

     Thus, the Great theorem of Fermat in the expanded kind it is possible, in view of the accepted assumption about existence of rectangular triangles, it is possible to formulate:

     For any Real number 1<n≠2 equation xⁿ + yⁿ = zⁿ has no decisions in Real numbers nonzero x, y, z if it cannot be transformed by simplification to the equation which is not adequating to given conditions.