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

2024/12/12 by Deshouillers, Jean-Marc, Jelinek, Pascal, Spiegelhofer, Lukas
#11A63 #11B75 #11K31 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.09124

Abstract

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many collisions of the base-2 and base-3 sum-of-digits functions. In other words, the equation s₂(n)=s₃(n) admits infinitely many solutions in natural numbers. We refine this result and prove that every integer a in \1, 2, …, 11\ appears as the last nonzero digit of n! in base 12 infinitely often.

Related

• Collisions of digit sums in bases 2 and 3
• The joint distribution of binary and ternary digits sums
• On the binary digits of n and n²
• The binary digits of n+t
• On the last nonzero digits of n! in a given base