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discovered formula (2) after transforming the Pythagoras’ equation into an algebraic quadratic equation – see Appendix IV story Year 1652. However, an algebraic solution does not give an understanding the essence of the resulting formula. This method was first published in 2008 [30].

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For example, if m = p₁p₂ then in addition to the first three solutions there will be others: A₄=p₁; B₄=2p₁p₂²; A₅=p₂; B₅=2p₁²p₂; A₆=2p₁; B₆=p₁p₂²; A₇=2p₂; B₇=p₂p₁²; A₈=p₁²; B₈=2p₂²; A₉=p₂²; B₉=2p₁²

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Formula (7) is called Fermat Binomial. It is curious that the same name appeared in 1984 in the novel "Sharper than the epee" by the Soviet science fiction writer Alexander Kazantsev. This formula is not an identity because in contrast to the identity of Newton Binomial in addition to summands, there is also a sum of them, but with the help of Fermat Binominal it is easy to derive many useful identities in particular, factorization of the sum and difference of two identical powers [30], see also Pt. 4.4.

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In this case, identity (9) indicates that the same key formula is substituted into the transformed key formula (2) or that the equation (8) we obtained, is a key formula (2) in power n. But you can go the reverse way just give the identity (9) and then divide into factor the differences of powers and such a way you can obtain (8) without using the Fermat Binominal (7). But this way can be a trick to hide the understanding of the essence because when some identity falls from the sky, it may seem that there is nothing to object. However, if you memorize only this path, there is a risk of exposure in a misunderstanding of the essence because the question how to obtain this identity, may go unanswered.

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Taking into account that c−a=b−2m the expression in square brackets of equation (8) can be transformed as follows: (c++b)_n − (a++2m)_n = с^n-1− a^n-1+ c^n-2b− a^n-22m+ c^n-3b²− a^n-3(2m)²+ … +b^n-1 − (2m)^n-1; с^n-1 − a^n-1 = (с − a)(c++a)_n-1; c^n-2b − a^n-22m = 2m(c^n-2 − a^n-2) + c^n-2(b − 2m) = (c − a)[2m(c++a)_n-2 + c^n-2]; c^n-3b² −a^n-3(2m)² = (2m)²(c^n-3 − a^n-3) + c^n-3(b² − 4m²) = (c − a)[4m²(c++a)_n-3 + c^n-3(b +2m)]; b^n-1 − (2m)^n-1 = (b − 2m)(b++2m)_n-1 = (c − a)(b++2m) _n-1 All differences of numbers except the first and last, can be set in general form: c^xb^y − a^x(2m)^y=(2m)^y(c^x − a^x) + c^x[b^y − (2m)^y] = (c − a)(c++a)_x(2m)^y + (b − 2m)(b++2m)_yc^x = (c − a)[(c++a)_x(2m)^y + (b++2m)_yc^x] And from here it is already become clear how the number (c − a) is take out of brackets. Similarly, you can take out of brackets the factor a + b = c + 2m. But this is possible only for odd powers n. In this case, equation (10) will have the form A_iB_iC_iD_i = (2m)ⁿ, where A_i = c – b = a − 2m; B_i = c – a = b − 2m; C_i = a + b = c + 2m; D_i – polynomial of power n − 3 [30].

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Equation (10) can exist only if (1) holds i.e. {aⁿ+bⁿ−cⁿ}=0 therefore, any option with no solutions leads to the disappearance of this ghost equation. And in particular, there is no “refutation” that it is wrong to seek a solution for any combination of factors, since A_iB_i=2m² may contradict E_i=2^n-1m^n-2, when equating E_i to an integer does not always give integer solutions because a polynomial of power n−2 (remaining after take out the factor c−a) in this case may not consist only of integers. However, this argument does not refute the conclusion made, but rather strengthens it with another contradiction because E_i consists of the same numbers (a, b, c, m) as A_i, B_i where there can be only integers.

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In this proof, it was quite logical to indicate such a combination of factors in equation (10), from which the Pythagoras’ numbers follow. However, there are many other possibilities to get the same conclusion from this equation. For example, in [30] a whole ten different options are given and if desired, you can find even more. It is easy to show that Fermat's equation (1) is also impossible for fractional rational numbers since in this case, they can be led to a common denominator, which can then be reduced. Then we get the case of solving the Fermat equation in integers, but it has already been proven that this is impossible. In this proof of the FLT new discoveries are used, which are not known to current science: there are the key formula (2), a new way to solve the Pythagoras’ equation (4), (5), (6), and the Fermat Binomial formula (7) … yes of course, else also magic numbers from Pt. 4.3!!!