Fungrim home page

Fungrim entry: 540931

gp={10,p=404877,p=6692367337min(A),otherwise   where A={a:aZ1and#{akmodp:kZ0}=p1}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:pPandp3andp<1012p \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
ConreyGeneratorgpg_{p} Conrey generator
MinimumminP(x)f ⁣(x)\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right) Minimum value of a set or function
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
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:
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, SetBuilder(a, a, And(Element(a, ZZGreaterEqual(1)), Equal(Cardinality(SetBuilder(Mod(Pow(a, k), p), 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

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

2019-08-19 14:38:23.809000 UTC