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Adic Topologies for the Rational Integers


Published:20030801
Printed: Aug 2003
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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, quasivaluation, topological ring, completion, inverse limit, diophantine equation, prime integers, Fermat numbers, Fibonacci numbers
$p$adic, metrizable, quasivaluation, topological ring, completion, inverse limit, diophantine equation, prime integers, Fermat numbers, Fibonacci numbers

MSC Classifications: 
11B05, 11B25, 11B50, 13J10, 13B35 show english descriptions
Density, gaps, topology Arithmetic progressions [See also 11N13] Sequences (mod $m$) Complete rings, completion [See also 13B35] Completion [See also 13J10]
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]
