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

Theorem
Let $p$ be a prime number.

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

Also see



 * Construction of Regular Heptadecagon

Historical Note
This result was stated, but not proved, by, who demonstrated the result for $n = 17$ in 1796.

The case $p = 257$ was demonstrated by in 1832.

The case $p = 65, 537$ was attempted by, 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.