On the base $b$ expansion of the number of trailing zeroes of $b^k!$

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 the connection between the expression of $\vartheta (b):={\lim }_{n\to \infty }{Z}_b(n)/n$ in base $b$, and that of $Z_b(b^k)$. In particular, if $b$ is a prime power, we will show the equality between the $k$ digits of $Z_b(b^k)$ and the first $k$ digits in the fractional part of $\vartheta (b)$. In the general case we will see that this equality still holds except for, at most, the last $\lfloor {\log }_b(k)+3\rfloor$ digits. We finally show that this bound can be improved if $b$ is square-free and present some conjectures about this bound.

References in corpus (1)

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