From: Martin GOLDSTERN Newsgroups: sci.math.research Subject: Re: The Book Date: 17 Sep 1998 13:35:08 GMT Tim Chow tchow AT alum DOT mit DOT edu wrote: : Having said all this, I think the original question was a good one. Does : anyone have their own favorite examples of beautiful proofs that (1) are : deeper than the standard elementary proofs that everyone knows yet (2) are : accessible to a general audience of research mathematicians? There is a recent book by Martin Aigner and Guenter Ziegler called "Proofs from The Book". (Springer, ISBN 3-540-63698-6) The preface states "to a large extent this book reflects the views of Paul Erdos". Also "everything in this book is supposed to be accessible to readers [knowing only some] undergraduate mathematics". It contains 6 proofs of the infinitude of primes, starting with Euclid's of course. One proof shows that the Fermat numbers F_n=2^2^n+1 must all be relatively prime [ use F_n*(F_n-2)=F_{n+1}-2 ]. Another uses Lagrange's theorem (element order divides group order) to show that for any prime factor q of 2^p-1 we must have p|q-1, so p 0. Since there are only finitely many remainders mod b , one can see that these basic neighborhoods are not only open but also closed. Now look at the union of all sets N_{0,p}, over all primes p. If there were only finitely many primes, this set would also be closed, but its complement is just {1,-1} (since all other numbers have some prime divisor), and {1,-1} is certainly not open (all nonempty open sets are infinite) Neat, isn't it? Martin.Goldstern@tuwien.ac.at ============================================================================== From: propp@math.mit.edu (Jim Propp) Newsgroups: sci.math.research Subject: Re: The Book Date: 16 Sep 1998 20:02:34 -0400 Here's that proof of the uniqueness of factorization in Z that I mentioned, courtesy of someone who prefers to be nameless: Suppose N = p_1...p_r = q_1...q_s is the smallest positive integer with two different factorizations. Then no p_i equals any q_j. WLOG, suppose p_1 < q_1. Let N' = (q_1-p_1)q_2...q_s. Factor q_1-p_1 into primes to turn this into a prime factorization of N' not including the factor p_1 (p_1 does not divide q_1-p_1 since p_1 does not divide q_1). However, writing N' = p_1(p_2...p_r-q_2...q_s) leads to a prime factorization containing p_1. This contradicts the minimality of N, since 0 < N' < N. Jim Propp