65537-gon

65537 is the largest known Fermat prime, and the 65537-gon is therefore a constructible polygon using compass and straightedge, as proved by Gauss. The 65537-gon has so many sides that it is, for all intents and purposes, indistinguishable from a circle using any reasonable printing or display methods.

The values and are algebraic numbers of degree 32768.

Hermes spent 10 years on the construction of the 65537-gon at Königsberg around 1900. After the Second World War, his manuscripts were moved to the Mathematical Institute in Göttingen, where they can now be viewed (Coxeter 1969).

DeTemple (1991) notes that a geometric construction can be done using 1332 or fewer Carlyle circles.