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Fungrim entry: 75231e

gp=min{a:aZ1  and  #{akmodp:kZ0}=p1  and  #{akmodp2:kZ0}=p(p1)}g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}
Assumptions:pP  and  p3p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3
g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3
Fungrim symbol Notation Short description
ConreyGeneratorgpg_{p} Conrey generator
MinimumminxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Cardinality#S\# S Set cardinality
Powab{a}^{b} Power
PPP\mathbb{P} Prime numbers
Source code for this entry:
    Formula(Equal(ConreyGenerator(p), Minimum(Set(a, For(a), And(Element(a, ZZGreaterEqual(1)), Equal(Cardinality(Set(Mod(Pow(a, k), p), For(k), Element(k, ZZGreaterEqual(0)))), Sub(p, 1)), Equal(Cardinality(Set(Mod(Pow(a, k), Pow(p, 2)), For(k), Element(k, ZZGreaterEqual(0)))), Mul(p, Sub(p, 1)))))))),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC