From: tzs@stein2.u.washington.edu (Tim Smith)
Newsgroups: sci.math
Subject: Re: Dirichlet's theorem
Date: 5 Jul 93 00:12:32 GMT
goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard) writes:
>Well,...there's a "dumb" proof that there are infinitely many
>primes by showing that the sum of the reciprocals of the primes
>diverges. (Dumb, since, by the time you can understand the
>divergence of the sum, you already know several easier proofs.)
You would probably like Furstenberg's proof, which Ribenboim quotes
in "The Book of Prime Number Records":
In this note we would like to offer an elementary "topological"
proof of the infinitude of the prime numbers. We introduce a
topology into the space of integers S, by using the arithmetic
progressions (from -oo to +oo) as a basis. It is not difficult
to verify that this actually yields a topological space. In fact,
under this topology, S may be shown to be normal and hence
metrizable. Each arithmetic progression is closed as well as
open, since its complement is the union of the other arithmetic
progressions (having the same difference). As a result, the union
of any finite number of arithmetic progressions is closed.
Consider now the set A = U Ap, where Ap consists of all
multiples of p, and p runs through the set of primes >= 2. The
only numbers not belonging to A are -1 and 1, and since the
set {-1, 1} is clearly not an open set, A cannot be closed. Hence,
A is not a finite union of closed sets which proves that there
are an infinity of primes.
--
"Pope moved that we strike from the State's brief and appendix a selection from
the Year Book of 1484 written in Medieval Latin and references thereto. The
State provided no translation and conceded a total lack of knowledge of what it
meant. The motion is granted" 396 A.2d 1054 --Tim Smith