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Totative

Coprime number less than a given integer

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated November 15, 2025

In number theory, a totative of a given positive integer n is an integer k such that 0 < k ≤ n and k is coprime to n. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.

01Distribution

The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as

0<a_{1}<a_{2}\cdots <a_{\phi (n)}<n,

the mean square gap satisfies

\sum _{i=1}^{\phi (n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi (n)

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.

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This article is adapted from the Wikipedia article “Totative”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Reduced residue system

In mathematics, a subset R of the integers is called a reduced residue system modulo n if: gcd = 1 for each r in R, R contains φ(n) elements, no two elements of R are congruent modulo n. Here φ denotes Euler's totient function.