  We introduce a collection of polynomials $F_N$, associated to each positive
integer $N$, whose divisibility properties yield a reformulation of the
Goldbach conjecture. While this reformulation certainly does not lead to a
resolution of the conjecture, it does suggest two natural generalizations for
which we provide some numerical evidence. As these polynomials $F_N$ are
independently interesting, we further explore their basic properties, giving,
among other things, asymptotic estimates on the growth of their coefficients.
