Fermat's Last Theorem (3 Lines Way)

Abstract 〖 is a〗^n+b^n=〖 c〗^n? Where a,b,c,n∈N_+,n>2 (*) Binomial Theorem The binomial theorem〖 (x+y)〗^n guarantees that every integer n∈ N_+ yields a non-zero integer, thereby preserving the intermediate terms; these terms remain part of the final polynomial less than〖 (x+y)〗^n. This paper presents a direct algebraic proof of Fermat’s Last Theorem. By examining the linear relationship between the bases a, b, and c, we partition the problem into two distinct cases. The trivial case, where a + b ≤ c, is resolved directly using strict power inequalities to get〖 a〗^n 〖+b〗^n≠c^n . For the non-trivial case, where a + b > c, we introduce a shifting framework using positive integer parameters u and v to construct an identity based on the Binomial Theorem. We prove that the resulting cross-term remainder function h is strictly positive (h ∈N_+). This logical way establishes a^n 〖+b〗^n≠c^n