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# Def/Totient

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Definition of Totient: Suppose that $m$ is a positive integer. Let $\Phi(m)$ denote the set of invertible residues, modulo $m$. Euler's totient of $m$, denoted $\phi(m)$ is defined to be the cardinality of the finite set $\Phi(m)$.

For example, in the case when $m = 12$, $$\Phi(12) = \{ \bar 1, \bar 5, \bar 7, \bar 11 \}.$$ Hence, $\phi(12) = \vert \Phi(12) \vert = 4$. There are four invertible residues, modulo $12$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Invertible residue, Def/Cardinality

The following statements and definitions logically rely on the material of this page: State/Computing the totient of a prime power, State/Fermat Euler theorem, and State/Multiplicativity of the totient

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Modular arithmetic