Fungrim entry: a68214

{a}^{\varphi(n)} \equiv 1 \pmod {n}

Assumptions:

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, n\right) = 1

TeX:

{a}^{\varphi(n)} \equiv 1 \pmod {n}

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, n\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Totient`	$\varphi(n)$	Euler totient function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("a68214"),
    Formula(CongruentMod(Pow(a, Totient(n)), 1, n)),
    Variables(a, n),
    Assumptions(And(Element(a, ZZ), Element(n, ZZGreaterEqual(1)), Equal(GCD(a, n), 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