The eighteen year old Gauss began his scientific diary with his construction of the regular 17-gon. The Greeks had ruler-and-compass constructions for the regular polygons with 3, 4, 5, and 15 sides, and for all others obtainable from these by doubling the number of sides. Gauss completely solved the problem by proving that: A regular n-gon is constructible if n is a product of a power of 2 and at most two distinct Fermat primes. This discovery led Gauss to devote his life to mathematics rather than philology.
More information about:
Carl Friedrich Gauss
Constructable Polygons and Gauss's 17-gon