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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(n) \right\} \le 1.328 \sqrt{n \log(n)}
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(n) \right\} \le 1.328 \sqrt{n \log(n)}

n \in \mathbb{Z}_{\ge 5}
Definitions:
Fungrim symbol Notation Short description
MaximummaxxSf(x)\mathop{\max}\limits_{x \in S} f(x) Maximum value of a set or function
PPP\mathbb{P} Prime numbers
LandauGg(n)g(n) Landau's function
Sqrtz\sqrt{z} Principal square root
Loglog(z)\log(z) 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(Set(p, For(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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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC