Fungrim entry: 540931

g_{p} = \begin{cases} 10, & p = 40487\\7, & p = 6692367337\\\min\left(A\right), & \text{otherwise}\\ \end{cases}\; \text{ where } A = \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 \right\}

Assumptions:

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; p < {10}^{12}

TeX:

g_{p} = \begin{cases} 10, & p = 40487\\7, & p = 6692367337\\\min\left(A\right), & \text{otherwise}\\ \end{cases}\; \text{ where } A = \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 \right\}

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; p < {10}^{12}

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("540931"),
    Formula(Where(Equal(ConreyGenerator(p), Cases(Tuple(10, Equal(p, 40487)), Tuple(7, Equal(p, 6692367337)), Tuple(Minimum(A), Otherwise))), Equal(A, 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))))))),
    Variables(p),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3), Less(p, Pow(10, 12)))))

Topics using this entry

Dirichlet characters