Fermat' s Last Theorem (Best Way)

<jats:p>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 \(n &gt; 2\). 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 (\(k &gt; h\)) 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.</jats:p>