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 Adic Topologies for the Rational Integers 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 numbersCategories:11B05, 11B25, 11B50, 13J10, 13B35