Fungrim entry: 87d19b

\max \left\{ p : p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \mid g(n) \right\} \le 1.328 \sqrt{n \log(n)}

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

n \in \mathbb{Z}_{\ge 5}

Definitions:

Fungrim symbol	Notation	Short description
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`PP`	$\mathbb{P}$	Prime numbers
`LandauG`	$g(n)$	Landau's function
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	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(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"))

Topics using this entry

Landau's function