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Fungrim entry: 36fe36

(xy(modφ ⁣(n)))    (axay(modn))\left(x \equiv y \pmod {\varphi\!\left(n\right)}\right) \implies \left({a}^{x} \equiv {a}^{y} \pmod {n}\right)
Assumptions:aZandnZ1andgcd ⁣(a,n)=1andxZ0andyZ0a \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\!\left(n\right)}\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
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
GCDgcd ⁣(a,b)\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)))))

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2019-08-17 11:32:46.829430 UTC