The problem of which

regular polygons can be constructed
by

ruler and compass alone goes back to the ancient Greeks.
They could certainly construct a regular

pentagon, for
example. Fast forward a couple of thousand years
and the nineteen year-old

*wunderkind* Gauss
made the first new
progress on the problem when he constructed the
regular 17-gon. Gauss discovered necessary
and sufficient conditions for constructibility of
the regular

*n*-gon and he recorded his
result in Section VII of

*Disquisitiones
Arithemeticae* published in 1801. To
explain Gauss's result we need a definition.

**Definition** A number of the form
*2*^{2n}+1 with *n*
a nonnegative integer is called a
*Fermat number*.

The first 5 Fermat numbers are:
3, 5, 17, 257, 65537. All of these are prime
and Fermat conjectured in 1640 that all
the Fermat numbers are prime. Rather surprisingly
it wasn't until 1732 that Euler pointed out that the
next Fermat number 4294967297 is not prime.
It is divisible by 641. In fact the first 5 are the only
know prime Fermat numbers and it seems reasonable
that there are no others.

**Theorem** (Gauss) A regular *n*-gon is
constructible by ruler and compass alone
if and only if

*
n = 2*^{k}p_{1}...p_{t}

where

*k* and

*t* are nonnegative integers
and

*p*_{i} are distinct prime Fermat
numbers.