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Fungrim entry: b9c36d

φ ⁣(k=1mpkek)=k=1mφ ⁣(pkek)\varphi\!\left(\prod_{k=1}^{m} p_{k}^{{e}_{k}}\right) = \prod_{k=1}^{m} \varphi\!\left(p_{k}^{{e}_{k}}\right)
Assumptions:ekZ0andmZ0{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
Totientφ(n)\varphi(n) Euler totient function
Productnf(n)\prod_{n} f(n) Product
Powab{a}^{b} Power
PrimeNumberpnp_{n} nth prime number
ZZGreaterEqualZn\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)))))

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

2019-10-05 13:11:19.856591 UTC