# Fungrim entry: 36fe36

$\left(x \equiv y \pmod {\varphi(n)}\right) \;\implies\; \left({a}^{x} \equiv {a}^{y} \pmod {n}\right)$
Assumptions:$a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, n\right) = 1 \;\mathbin{\operatorname{and}}\; x \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; y \in \mathbb{Z}_{\ge 0}$
TeX:
\left(x \equiv y \pmod {\varphi(n)}\right) \;\implies\; \left({a}^{x} \equiv {a}^{y} \pmod {n}\right)

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, n\right) = 1 \;\mathbin{\operatorname{and}}\; x \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; y \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Totient$\varphi(n)$ Euler totient function
Pow${a}^{b}$ Power
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("36fe36"),
Formula(Implies(CongruentMod(x, y, Totient(n)), CongruentMod(Pow(a, x), Pow(a, y), n))),
Variables(a, x, y, n),
Assumptions(And(Element(a, ZZ), Element(n, ZZGreaterEqual(1)), Equal(GCD(a, n), 1), Element(x, ZZGreaterEqual(0)), Element(y, 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.

2021-03-15 19:12:00.328586 UTC