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 in the variable . 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 of the equation f(k, p)=0 and the number corresponding to this supposed solution 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
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