  Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and
$Z_n$ with the property that $\mu$ is a zero of $X_n$ if and only if the triple
$(\mu,\mu+m,\mu+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the
irreducibility of these polynomials implies Fermat's last theorem. It is also
shown, in a precise asymptotic sense, that for a vast majority of cases, these
polynomials are irreducible via Eisenstein's criterion. We conclude by offering
a conjecture on powerful numbers.
