A new look at the trailing zeroes on $N!$

TL;DR

This paper investigates the behavior of the function counting trailing zeros of factorials in various bases, characterizing where it increases, its jump amplitudes, and exploring its asymptotic properties and image gaps.

Contribution

It provides a detailed analysis of the function $Z_b(n)$, including its increase points, jump sizes, asymptotic behavior, and the characterization of integers not in its image.

Findings

Identifies points where $Z_b(n)$ increases and computes jump amplitudes.

Studies asymptotic behavior of $Z_b(n)$.

Characterizes integers not in the image of $Z_b$.

Abstract

Let us denote by $Z_b(n)$ the number of trailing zeroes in the base $b$ expansion of $n!$. In this paper we study with some detail the behavior of the function $Z_b$. In particular, since $Z_b$ is non-decreasing, we will characterize the points where it increases and we will compute the amplitude of the jump in each of such points. In passing, we will study some asymptotic aspects and we will give families of integers that do not belong to the image of $Z_b$.