# Fungrim entry: 75231e

$g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}$
Assumptions:$p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3$
TeX:
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
Definitions:
Fungrim symbol Notation Short description
ConreyGenerator$g_{p}$ Conrey generator
Minimum$\mathop{\min}\limits_{x \in S} f(x)$ Minimum value of a set or function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Cardinality$\# S$ Set cardinality
Pow${a}^{b}$ Power
PP$\mathbb{P}$ Prime numbers
Source code for this entry:
Entry(ID("75231e"),
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)))))))),
Variables(p),
Assumptions(And(Element(p, PP), GreaterEqual(p, 3))))

## 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