On a problem of Chen and Liu concerning the prime power factorization of $n!$

TL;DR

This paper investigates the distribution of prime power exponents in factorials, providing answers to specific modular questions and asymptotic formulas for their counts in arithmetic progressions, extending to higher powers of integers.

Contribution

It answers open questions by Chen and Liu regarding the distribution of prime power exponents in factorials and derives asymptotic formulas for their counts in arithmetic progressions.

Findings

Confirmed that the sets of exponents modulo m are complete for squares and primes.

Derived asymptotic formulas for the number of n and q satisfying certain modular conditions.

Established lower bounds for exponents of higher powers in factorials.

Abstract

For a fixed prime $p$, let $e_p(n!)$ denote the order of $p$ in the prime factorization of $n!$. Chen and Liu (2007) asked whether for any fixed $m$, one has $\{e_p(n^2!) \bmod m:\; n\in\mathbb{Z}\}=\mathbb{Z}_m$ and $\{e_p(q!) \bmod m:\; q {prime}\}=\mathbb{Z}_m$. We answer these two questions and show asymptotic formulas for $# \{n<x: n \equiv a \bmod d,\; e_p(n^2!)\equiv r \bmod m\}$ and $# \{q<x: q {prime}, q \equiv a \bmod d,\; e_p(q!)\equiv r \bmod m\}$. Furthermore, we show that for each $h\geq 3$, we have $\{n<x: n \equiv a \bmod d,\; e_p(n^h!)\equiv r \bmod m\} \gg x^{4/(3h+1)}$.