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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

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