Fungrim entry: 56d7fe

\varphi\!\left(m n\right) = \frac{\varphi(m) \varphi(n) \gcd\!\left(m, n\right)}{\varphi\!\left(\gcd\!\left(m, n\right)\right)}

Assumptions:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\varphi\!\left(m n\right) = \frac{\varphi(m) \varphi(n) \gcd\!\left(m, n\right)}{\varphi\!\left(\gcd\!\left(m, n\right)\right)}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Totient`	$\varphi(n)$	Euler totient function
`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("56d7fe"),
    Formula(Equal(Totient(Mul(m, n)), Div(Mul(Mul(Totient(m), Totient(n)), GCD(m, n)), Totient(GCD(m, n))))),
    Variables(m, n),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(n, ZZGreaterEqual(1)))))

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