Fungrim entry: c19cd6

\varphi(n) = \# \left\{ k : k \in \{1, 2, \ldots, n\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1 \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\varphi(n) = \# \left\{ k : k \in \{1, 2, \ldots, n\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1 \right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Totient`	$\varphi(n)$	Euler totient function
`Cardinality`	$\# S$	Set cardinality
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("c19cd6"),
    Formula(Equal(Totient(n), Cardinality(Set(k, ForElement(k, Range(1, n)), Equal(GCD(n, k), 1))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

Totient function

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC