From: utzoo!decvax!ucbvax!G:shallit
Newsgroups: net.math
Title: Re: pizza
Article-I.D.: populi.213
Posted: Tue Jun 22 10:21:45 1982
Received: Sun Jun 27 02:11:23 1982

Why you would want to explain it in plain English is beyond me.
Mathematical notation is so much more precise and clear for this
kind of thing.

So, after starting thus, here goes an attempt at an explanation.  It
has absolutely nothing to do with the prime factorization of 360, as
was suggested.  (How could it?  360 is an artificial number, decided
by man, not nature.  If we decided on 420 degrees in a circle, would
this make it possible to divide a circle into 7 parts?  Of course not.
But I digress.)

Gauss showed that the circle can be divided into n parts if and
only if n is 1) a Fermat prime (i. e. of the form 1+2*(2*k) ) or
(2) a product of such primes or (3) the product of a power of
2 and a set of Fermat primes.

His reasoning was approximately as follows:  A quantity can be
constructed by straightedge and compass if it can be expressed
by a finite number of applications of + - * / sqrt.

There are polynomials all of whose roots can be expressed in this
way; among such, there is one of minimal degree with a given
quantity as a root.  The degree of these polynomials is a power of 2.

The roots of x^n - 1 = 0 lie on a circle , spaced equally around
the circumference.  In order to divide the circle we must be able
to construct these points.  We can assume n is a prime or a power
of a prime, since if we can construct the roots of x^p - 1 = 0 and
x^q - 1 = 0 we can construct the roots of x^pq - 1 = 0.

We can certainly bisect an arc with straightedge and compass.  Hence
we can divide a circle into 2, 4, 8 , ... parts.  If we could
divide a circle into k parts (k odd) we could divide it into
(2^j)*k parts.

Look at x^p - 1 = 0, where p is prime.  Divide this by x-1 to
get an irreducible (unfactorable) equation of degree p-1.  Since
it is irreducible, it is of minimal degree and so p-1 must
be a power of 2.  The only numbers p-1 that are powers of 2 are
such that p = 1+2^2^n, a Fermat prime.  The only ones known
are 3,5, 17, 257, and 65537.

Hope this helps a little.