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Fungrim entry: 87d19b

max{p:pPandpg ⁣(n)}1.328nlog ⁣(n)\max \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \mid g\!\left(n\right) \right\} \le 1.328 \sqrt{n \log\!\left(n\right)}
Assumptions:nZ5n \in \mathbb{Z}_{\ge 5}
References:
  • Jon Grantham (1995), The largest prime dividing the maximal order of an element of S_n, 64, 209, pp. 407--210, https://doi.org/10.2307/2153344
TeX:
\max \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \mid g\!\left(n\right) \right\} \le 1.328 \sqrt{n \log\!\left(n\right)}

n \in \mathbb{Z}_{\ge 5}
Definitions:
Fungrim symbol Notation Short description
MaximummaxP(x)f ⁣(x)\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right) Maximum value of a set or function
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
PPP\mathbb{P} Prime numbers
LandauGg ⁣(n)g\!\left(n\right) Landau's function
Sqrtz\sqrt{z} Principal square root
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("87d19b"),
    Formula(LessEqual(Maximum(SetBuilder(p, p, And(Element(p, PP), Divides(p, LandauG(n))))), Mul(Decimal("1.328"), Sqrt(Mul(n, Log(n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(5))),
    References("Jon Grantham (1995), The largest prime dividing the maximal order of an element of S_n, 64, 209, pp. 407--210, https://doi.org/10.2307/2153344"))

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2019-08-21 11:44:15.926409 UTC