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

Totient function