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