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# Adic Topologies for the Rational Integers

Published:2003-08-01
Printed: Aug 2003
• Kevin A. Broughan
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## Abstract

A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.
 Keywords: $p$-adic, metrizable, quasi-valuation, topological ring, completion, inverse limit, diophantine equation, prime integers, Fermat numbers, Fibonacci numbers
 MSC Classifications: 11B05 - Density, gaps, topology 11B25 - Arithmetic progressions [See also 11N13] 11B50 - Sequences (mod $m$) 13J10 - Complete rings, completion [See also 13B35] 13B35 - Completion [See also 13J10]