# Fungrim entry: 87d19b

$\max \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \mid g\!\left(n\right) \right\} \le 1.328 \sqrt{n \log\!\left(n\right)}$
Assumptions:$n \in \mathbb{Z}_{\ge 5}$
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
Maximum$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$ Maximum value of a set or function
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
PP$\mathbb{P}$ Prime numbers
LandauG$g\!\left(n\right)$ Landau's function
Sqrt$\sqrt{z}$ Principal square root
Log$\log\!\left(z\right)$ Natural logarithm
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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"))

## Topics using this entry

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

2019-08-21 11:44:15.926409 UTC