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Definition: Subsets of Prime Numbers Not Dividing a Natural Number

Let \(d\ge 0\) be a natural number. By \(\mathbb P_d\) we denote the subset of prime numbers \(\mathbb P\) not dividing \(d\). Formally

\[p\in \mathbb P_d\Longleftrightarrow p\not\mid d.\]

Examples

• \(\mathbb P_0=\emptyset\), since all prime numbers divide \(0\) and therefore none do not divide \(0\).
• \(\mathbb P_1=\mathbb P\), since all prime numbers do not divide \(1\),
• \(\mathbb P_p=\mathbb P\setminus\{p\}\) for any prime \(p\), since all prime numbers except \(p\) do not divide \(p\),
• \(\mathbb P_d=\mathbb P\setminus\{p:~p\mid d\}\) for any prime \(p\) which divides the number \(d\).

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References

Bibliography

1. Piotrowski, Andreas: "Anmerkungen zur Verteilung der Primzahlzwillinge", Master’s thesis, Frankfurt am Main, 1999