Regular Heptagon is Smallest with no Compass and Straightedge Construction/Proof 2

Theorem

The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction.

Proof

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A theorem states that: "The n-gon is constructible by compass and straightedge construction if and only if $n = 2^kn_0$, with $k \in \mathbb{Z}_{\geq0}$ and $n_0$ the product of any number of distinct Fermat primes."

Note that 3 and 5 are both Fermat primes, so their respective n-gon are both constructible; and $4$ is just a power of $2$, so the 4-gon also constructible.

Lastly, since 6 is the product of 2 and 3, the 6-gon is constructible as well. Hence, we remain with 7, which does not satisfy the theorem's criteria; and thus the 7-gon is the smallest regular polygon that cannot be constructed using a compass and straightedge construction.

$\blacksquare$