Construction of Regular Prime p-Gon Exists iff p is Fermat Prime

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Let $p$ be a prime number.

Then there exists a compass and straightedge construction for a regular $p$-gon if and only if $p$ is a Fermat prime.


Also see

Historical Note

The result Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime was stated, but not proved, by Carl Friedrich Gauss, who demonstrated the result for $n = 17$ in $1796$, when he was $18$.

The case $p = 257$ was demonstrated by Magnus Georg Paucker in $1822$, and again by Friedrich Julius Richelot in $1832$.

The case $p = 65 \, 537$ was attempted by Johann Gustav Hermes, who offered a construction in $1894$ after a decade of work. However, it has been suggested that there are mistakes in his work.

The cases where $p = 3$ and $p = 5$ were known to the ancient Greeks and are given in Euclid's The Elements.