On the prime power factorization of n!

Abstract

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\nmid m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are uniformly distributed modulo $(m_1,...,m_k)$, where $e_p(n)$ is the order of the prime $p$ in the factorization of $n!$. That implies one of Sander's conjecture from \cite{S}, for any set of odd primes. Berend \cite{B} asks to find the fastest growing function $f(x)$ so that for large $x$ and any given finite sequence ${\epsilon }_i\in \{0,1\},i\le f(x)$, there exists $n<x$ such that the congruences $e_{p_i}(n)\equiv {\epsilon }_i(\mathrm{mod}2)$ hold for all $i\le f(x)$. Here, $p_i$ is the $i$th prime number. In our second result, we are able to show that $f(x)$ can be taken to be at least $c_1(\log x/(\log \log x{)}^6{)}^{1/9}$, with some absolute constant $c_1$, provided that only the first odd prime numbers are involved.

7 pages; accepted Journal of Number Theory