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Adic Topologies for the Rational Integers

Published online by Cambridge University Press:  20 November 2018

Kevin A. Broughan
Kevin A. Broughan*
Affiliation:
University of Waikato, Hamilton, New Zealand e-mail: kab@waikato.ac.nz
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Abstract

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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

11B0511B2511B5013J1013B35p-adic, metrizablequasi-valuationtopological ringcompletioninverse limitdiophantine equationprime integersFermat numbersFibonacci numbers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 4 , 01 August 2003 , pp. 711 - 723
Copyright
Copyright © Canadian Mathematical Society 2003

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