From: utzoo!decvax!harpo!utah-cs!utah-gr!thomas
Newsgroups: net.math
Title: Re: pizza
Article-I.D.: utah-gr.451
Posted: Tue Jun 22 10:16:25 1982
Received: Sun Jun 27 02:07:10 1982
References: utah-cs.789

There is a general rule for ruler and compass constructions, which can
be stated as follows:
	When the solution x of the problem is real and can be found
	by rational operations and (not necessarily real) square roots
	from the given numbers (line segment lengths) a, b, ..., the
	number (segment length) x can be constructed using ruler and
	compass.

In the case of dividing a circle into n equal parts, the condition is that
the number of divisors of n (phi(n)) is a power of 2.  Now, n can be written
in the form (2^m)(p1^m1)(p2^m2)...(pl^ml), where the p's are prime.  Then
phi(n) = (2^(m-1))(pq^(m1-1))...(pl^(ml-1))(p1-1)...(pl-1).  Thus, m1 thru
ml must all be 1, and (pi-1) must be a power of 2.  Now, it turns out that
all primes which are one greater than a power of 2 are of the form
2^(2^r)+1 for some integer r.  (Note, that not all numbers of this form
are prime, however, for example, 2^2^5 is divisible by 641.)  So, a
circle may be divided into n pieces if n is a product of some power of 2
and primes of this sort (to a most the first power).  The first few
primes in this sequence are 3, 5, 17, 257, 65537.  So, if you and 16
other friends were sharing a pizza, you could divide it into equal
pieces using only a straightedge and a compass.  The result is due to
Gauss, my presentation is a paraphrase of the presentation in Modern
Algebra, Vol 1, van der Waerden, pp183-187.

=Spencer