Binary-ternary collisions and the last significant digit of $n!$ in base 12

TL;DR

This paper proves that each last nonzero digit of factorials in base 12 occurs infinitely often, refining previous results on sum-of-digits collisions in different bases and connecting to factorial digit patterns.

Contribution

It establishes that all digits from 1 to 11 appear infinitely often as the last nonzero digit of n! in base 12, extending earlier work on digit collisions in different bases.

Findings

All digits 1 through 11 appear infinitely often as last nonzero digit of n! in base 12.

Refinement of previous results on sum-of-digits collisions in bases 2 and 3.

Infinitely many solutions exist for the last nonzero digit pattern in factorials in base 12.

Abstract

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits infinitely many solutions in natural numbers. We refine this result and prove that every integer $a$ in $\{1, 2, \ldots, 11\}$ appears as the last nonzero digit of $n!$ in base $12$ infinitely often.