It is difficult to find a person among mathematicians who is not familiar with and does not study the solution of the equation x²+y²+z²=0, or x³+y³+z³=0 in integers. A generalization of this problem is Fermat's Last Theorem, which states that the equation xⁿ+yⁿ+zⁿ=0 has no solutions in integers x, y, z and xyz ≠ 0, n≥3 is a natural number. This theorem was formulated in 1637 by the great French mathematician and logician P. Fermat. And it was noticed that he found a truly amazing proof. However, P. Fermat did not leave a proof in writing, except for the case n = 4. In 1753, the famous mathematician L. Euler proved the case n = 3. In his justification, he used the theory of complex numbers. Then, famous mathematicians L. Dirichlet, A. Legendre (n = 5, 1825), A. Lebesgue, G. Lame (n = 7, 1839), E. Kummer (n ≤ 100, 1847) proved it, and finally, 350 years later, in 1995, A. Wiles proved it for all natural numbers n ≥ 3. However, all the listed proofs, with the exception of the case n=4, cannot be considered simple. Our goal is to prove Fermat's Great Theorem for n=3 in a way that is more understandable for schoolchildren. The proposed proof will certainly contribute to the development of logical thinking and interest in mathematical science in schoolchildren. Let us briefly dwell on the proof: the proof uses the method of contradiction and the following definitions. We will write a ≡ 3 if the integer a is divisible by 3 without remainder, otherwise - a ≢ 3, while a non-negative number p satisfying the conditions a ≡ 3p, a ≢ 3p+1 is called the exponent of the multiplicity of a by 3 and is denoted by p = ω (a). In the proof, we first verify that q≥2, where q= ω (x+y+z), then in the course of further research q=1. The resulting contradiction proves the theorem.
A. Srashidinov (Fri,) studied this question.