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Fungrim entry: 7b27cd

#{k:k{1,2,n}andgcd ⁣(n,k)=1}=φ ⁣(n)\# \left\{ k : k \in \{1, 2, \ldots n\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1 \right\} = \varphi\!\left(n\right)
Assumptions:nZ1n \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\!\left(n\right)

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Cardinality#S\# S Set cardinality
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("7b27cd"),
    Formula(Equal(Cardinality(SetBuilder(k, k, And(Element(k, ZZBetween(1, n)), Equal(GCD(n, k), 1)))), Totient(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

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

2019-08-17 11:32:46.829430 UTC