MathDB

The limit of the multiplicity of p [ILL 1971]

Source:

January 1, 2011

limitfloor functionlogarithmsnumber theory proposednumber theory

Problem Statement

Denote by

x_n(p)

the multiplicity of the prime

p

in the canonical representation of the number

n!

as a product of primes. Prove that

\frac{x_n(p)}{n}<\frac{1}{p-1}

and

\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}