On the finite factorial numbers

TL;DR

This paper introduces finite factorial (FF) and partially finite factorial (PFF) transcendental numbers, explores their properties, and demonstrates that rational numbers cannot be FF, analyzing their topological and measure-theoretic characteristics.

Contribution

It defines new classes of transcendental numbers based on $b$-ary expansions and investigates their fundamental properties and distribution.

Findings

Rational numbers are not finite factorial.

The collection of FF numbers has specific topological properties.

Numerical examples illustrate key results.

Abstract

We introduce two families of transcendental numbers which we call finite factorial (FF) and partially finite factorial (PFF) numbers respectively, with the former one being subfamily of the latter one. These numbers arise naturally from some transcendental criterion for real numbers via their $b$-ary expansions. We show that rational numbers (eventually periodic words) can not be finite factorial. Then we consider the geometric (topological) properties of the collection of all the FF numbers, including its countability, density and Hausdorff dimension. Some numerical examples are given to illustrate certain results in the work.