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

Dirichlet characters