Construction of Regular 65,537-Gon
From Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime it is known that this construction is possible.
It was proved by Carl Friedrich Gauss in $1801$ that the construction is possible.
The first actual construction of a regular $65,537$-gon was attempted by Johann Gustav Hermes, who calculated the $384$ required quadratic equations in $1894$ after a decade of work. However, it has been suggested that there are mistakes in his work.