Fermat' s Last Theorem (Best Way)

Abstract This paper presents an elementary algebraic approach attempting to prove Fermat's Last Theorem, which states that no three positive integers \(a\), \(b\), and \(c\) satisfy the equation \(a^n + b^n = c^n\) for any integer value of \(n > 2\). By segmenting the relationship between the bases into three distinct trichotomy cases—\(a+b=c\), \(a+bc\)—the author utilizes binomial expansions to isolate cross-terms and establish inequalities. While the first two cases successfully demonstrate that no solutions exist, the third case introduces strict inequality bounds (\(k > h\)) on the expansion remainders to force a contradiction. Analysis reveals that this boundary condition implicitly assumes the non-existence of a solution, resulting in circular logic that invalidates the final contradiction.