The subject of this paper is the Gauss-Wantzel Theorem that states which regular $n$-gons can be constructed using only straightedge and compass. Said property of natural numbers $n$ is first analyzed among their basic building blocks in the form of primes (denoted by $p$), for which the theorem determines that regular $p$-gon is constructible if and only if $p$ is a Fermat prime (denoted by $p_F$). In that regard, each of the authors provided the proof of one of the directions of the proposed equivalence: Gauss first developed an algorithm that allows us to eventually construct a regular $p_F$-gon for any Fermat prime $p_F$, whereas Wantzel proved that no regular $n$-gons others than the ones already suggested by Gauss could ever be constructed.
Using appropriate integer combinations, central angles of different $p_{F_i}$-gons can be added into central angle of a $\prod_i p_{F_i}$-gon, which can further be repeatedly divided into two by consecutive angle bisections. Hence we have an algorithm on how to construct a regular $c$-gon for composite numbers $c$ in the form $c=2^kp_{F_1}p_{F_2}\ldots p_{F_t}$. As with the construction of single $p$-gons, it turns out that the sufficient condition for the application of this particular algorithm for their composition already aligns with the necessary one, therefore making the aforementioned $c$-gons precisely the ones being constructible.
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