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Fungrim entry: 36fe36

(xy(modφ(n)))        (axay(modn))\left(x \equiv y \pmod {\varphi(n)}\right) \;\implies\; \left({a}^{x} \equiv {a}^{y} \pmod {n}\right)
Assumptions:aZ  and  nZ1  and  gcd ⁣(a,n)=1  and  xZ0  and  yZ0a \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}
\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}
Fungrim symbol Notation Short description
Totientφ(n)\varphi(n) 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:
    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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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