# Fungrim entry: b9c36d

$\varphi\!\left(\prod_{k=1}^{m} p_{k}^{{e}_{k}}\right) = \prod_{k=1}^{m} \varphi\!\left(p_{k}^{{e}_{k}}\right)$
Assumptions:${e}_{k} \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}$
TeX:
\varphi\!\left(\prod_{k=1}^{m} p_{k}^{{e}_{k}}\right) = \prod_{k=1}^{m} \varphi\!\left(p_{k}^{{e}_{k}}\right)

{e}_{k} \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Totient$\varphi(n)$ Euler totient function
Product$\prod_{n} f(n)$ Product
Pow${a}^{b}$ Power
PrimeNumber$p_{n}$ nth prime number
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("b9c36d"),
Formula(Equal(Totient(Product(Pow(PrimeNumber(k), Subscript(e, k)), For(k, 1, m))), Product(Totient(Pow(PrimeNumber(k), Subscript(e, k))), For(k, 1, m)))),
Variables(e, m),
Assumptions(And(Element(Subscript(e, k), ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

## Topics using this entry

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

2020-04-08 16:14:44.404316 UTC