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 
In 1630 the French mathematician  the
fan, the lawyer by a trade Pierre Fermat (16011665) 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. 605608). After his death in P.Fermat's
papers have found the proof only for a degree equal 4. ____________________________________________________________________________ The first variant of the proof In the rectangular triangle having the parties x, y, z_{1} (fig. 1), is carried out equality z_{1}^{2} = x^{2} + y^{2} (1)
fig.1 At an exponent n>2 z_{1}^{n} = (x^{2} + y^{2})^{n}^{/2} > x^{n} + y^{n} (2) It is obvious, that in the formula z^{n} = x^{n} + y^{n} (3) z > y ≥ x or z > x ≥ y Thus, it is possible to ascertain, that to equality z^{n} = x^{n} + y^{n} 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 < z_{1} (4) The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle z^{2} < x^{2} + y^{2} (5) To conditions (3) and (5) satisfies also the acuteangled 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 acuteangled triangle concerning the party z z^{2} = x^{2} + y^{2} – 2xycos z (7) Thence follows z^{n} = (x^{2} + y^{2} – 2xycos z)^{n}^{/2} (8) In result it is possible to write down z^{n} = x^{n} + y^{n} = (x^{2} + y^{2} – 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 = (x^{2} + y^{2}  z^{2})/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 equalsign between irrational and a rational number, for example, is put z = π/2 = 90^{0}. 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 equalsign 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 acuteangled 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 acuteangled 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 z^{n} which would satisfy to equality (9). z^{n} ≠ x^{n} + y^{n} = (x^{2} + y^{2} – 2xycos z)^{n}^{/2} at n>2 (11) ____________________________________________________________________________
The second variant of the proof In the rectangular triangle having the parties x, y, z_{1} (fig. 1), is carried out equality z_{1}^{2} = x^{2} + y^{2} (1)
fig.1 At an exponent n>2 z_{1}^{n} = (x^{2} + y^{2})^{n}^{/2} > x^{n} + y^{n} (2) It is obvious, that in the formula z^{n} = x^{n} + y^{n} (3) z > y ≥ x or z > x ≥ y Thus, it is possible to ascertain, that to equality z^{n} = x^{n} + y^{n} 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 < z_{1} (4) The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle z^{2} < x^{2} + y^{2} (5) To conditions (3) and (5) satisfies also the acuteangled 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 acuteangled triangle concerning the party z z^{2} = x^{2} + y^{2} – 2xycos z (7) Thence follows z^{n} = (x^{2} + y^{2} – 2xycos z)^{n}^{/2} (8) In result it is possible to write down z^{n} = x^{n} + y^{n} = (x^{2} + y^{2} – 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 z^{n}, x^{n} and y^{n} multiplication of length of each of the parties to factors z^{n1}, x^{n1} and y^{n1} accordingly. Its decision will look like z^{2n} = x^{2n} + y^{2n} (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 z^{n} < x^{n} + y^{n} (11) But this value z^{n}^{ }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 ^{ } z^{n} < x^{n} + y^{n} = (x^{2} + y^{2} – 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. ^{ }___________________________________________________________________________
Expansion of the Great theorem of Fermat Having assumed as a basis the first variant of the proof, we shall consider expression (11) z^{n} ≠ x^{n} + y^{n} = (x^{2} + y^{2} – 2xycos z)^{n}^{/2} in the field of values n≤2. At n=2 the acuteangled 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 z^{n} ≠ x^{n} + y^{n} при 1<n<2 (1)
fig.1 At an exponent n>2 z_{1}^{n} = (x^{2} + y^{2})^{n}^{/2} > x^{n} + y^{n} (2) It is obvious, that in the formula z^{n} = x^{n} + y^{n} (3) z > y ≥ x or z > x ≥ y Thus, it is possible to ascertain, that to equality z^{n} = x^{n} + y^{n} 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 < z_{1} (4) The hypotenuse of not connected a rectangular triangle does not adjoin to a cathetus. At not connected a rectangular triangle z^{2} < x^{2} + y^{2} (5) To conditions (3) and (5) satisfies also the acuteangled 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 acuteangled triangle concerning the party z z^{2} = x^{2} + y^{2} – 2xycos z (7) Thence follows z^{n} = (x^{2} + y^{2} – 2xycos z)^{n}^{/2} (8) In result it is possible to write down z^{n} = x^{n} + y^{n} = (x^{2} + y^{2} – 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 z^{n} ≠ x^{n} + y^{n} = (x^{2} + y^{2} – 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 x^{n} + y^{n} = z^{n} 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.
