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.