Fungrim entry: 7b27cd

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

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

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

n \in \mathbb{Z}_{\ge 1}

Definitions:

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

Source code for this entry:

Entry(ID("7b27cd"),
    Formula(Equal(Cardinality(Set(k, For(k), And(Element(k, Range(1, n)), Equal(GCD(n, k), 1)))), Totient(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

Greatest common divisor

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