**Authors:** Philip A. Bloom

An open problem is proving FLT simply for each integral $n>2$. Our proof of FLT is based on our algebraic identity, denoted, {for convenience}, as $r^n+s^n=t^n$. For $n\geq1$ we relate $r,s,t$, each a different function of variables comprising $r^n+s^n=t^n$, with $x,y,z$ for which $x^n+y^n=z^n$ holds. We infer as true by \emph{direct argument} (not BWOC), for any given $n>2$, that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}$. In addition, we show, for $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{Z},r^n+s^n=t^n\}=\varnothing$. Thus, for $n\in\mathbb{Z},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{Z},x^n+y^n=z^n\}=\varnothing$.

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