Gauss-Wantzel theorem ($N=2^a{p}_1\cdots p_r$)

For $N\ge 3$, a regular polygon with $N$ sides is a constructible regular polygon exactly when $N=2^a{p}_1\cdots p_r$, where $a$ is a nonnegative integer and the $p_j$ are distinct Fermat primes; the product may be empty. Thus seventeen and fifteen qualify, while nine does not. Carl Friedrich Gauss established sufficiency, and Pierre Wantzel proved necessity in 1837.^[ref]