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'totative'
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| Title of object: |
totative |
| Canonical Name: |
Totative |
| Type: |
Definition |
| Created on: |
2007-04-23 11:36:05 |
| Modified on: |
2007-04-24 10:50:56 |
| Classification: |
msc:11A25 |
Revision comment (for changes between this and next version):
| Changes for correction #11803 ('Catching on'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
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Content:
Given a positive integer $n$, an integer $0 < m < n$ is a {\em totative} of $n$ if $\gcd(m, n) = 1$. Put another way, all the smaller integers than $n$ that are coprime to $n$ are totatives of $n$. Or, given the set of all the integers from 1 to $n - 1$, removing the subset of restricted divisors $d$ of $n$ (where we restrict the divisors to the range $1 < d < n$) leaves the set of the totatives of $n$.
For example, the totatives of 21 are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19 and 20.
The count of totatives of $n$ is Euler's totient function $\phi(n)$. The word ``totative'' was coined by James Joseph Sylvester, who also coined ``totient'' (though despite occasional usage in some papers and books, the term ``totative'' has yet to catch on). |
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