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Definition: Sets of Integers Co-Prime To a Given Integer

Let \(d\in\mathbb Z\) be an integer. By \(\mathbb Z_d\subseteq \mathbb Z\) we denote the subset of integers relatively prime to \(d\). Formally

\[n\in \mathbb Z_d\Longleftrightarrow n\perp d.\]

Examples

• \(\mathbb Z_0=\{1\}\), since \(\gcd(n,0)=1\) only for \(n=1\).
• \(\mathbb Z_1=\mathbb Z\), since \(\gcd(n,1)=1\) for all \(n\in\mathbb Z\).
• \(\mathbb Z_2\) are all odd numbers, since \(\gcd(n,2)=1\) holds for odd numbers, only.
• \(\mathbb Z_p\), $p$ being a prime number \(p\) are all integers not divisible by this prime number, since only for those we have \(\gcd(n,p)=1.\)
• etc.

Table of Contents

1. Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed

Mentioned in:

Lemmas: 1
Proofs: 2

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References

Bibliography

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