Geometric approach to the proof of Fermat’s last theorem

Abstract

A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat’s equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat’s equation has no entire solutions for p > 2. The numbers a = k, b = k +m, c = k+n, where k, m, n are natural numbers satisfying the inequalities n > m, n < k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle. The proof in this case is carried out by introducing a new auxiliary function f(k,p) = kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree p in the variable k. The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p). Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution k of the equation f(k, p) = 0 and the number (n −m ) corresponding to this supposed solution k are considered. The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author’s works, in which some special cases of Fermat’s last theorem are proved.