Although mathematicians have been working on this problem in Euclid's time (300 B.C.E.), it was Gauss who discovered it's construction in 1796 when he was an eighteen-year-old. Another significance is it is by this discovery that Gauss decided to spend him life persuing mathematics.
The Actual Proof
Gauss proved that a regular n-gon could be geometrically constructed if the number of sides were a prime number of the form:
Numbers of this form are also known as the Fermat numbers, they are not necessarily prime. For exmaple:
Gauss' theorem showed that regular polygons could be geometrically constructed with these number of sides. A 257-gon was constructed in 1832. A manuscript, which took ten years to finish, on directions for the construction of a 65,537-gon occupies a large box at the University of G�ttingen.
Also, another little known fact about this finding:
Gauss had always considered this finding one of his greatest, and told bolyai that this polygon would adorn his tombstone. For some reason, this was unable to be carried out, but the design is on the side of the base of a momument of him in Brunswick.
It is actually a seventeen-pointed star, the stonemason Howaldt said that the polygon would be mistaken as a circle.
