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On the finite factorial numbers

Liangang Ma

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.

On the finite factorial numbers

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 -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.
Paper Structure (7 sections, 26 theorems, 117 equations)

This paper contains 7 sections, 26 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Concepts and notations
  3. A transcendental criterion via the $b$-ary numerical system
  4. Applications of the criterion to reals with explicit expansions
  5. Eventually periodic words are not finite factorial
  6. The set of finite factorial numbers
  7. Numerical exhibitions

Key Result

Theorem 1.1

For an integer base $b\geq 2$ and an irrational real number $0<\xi<1$ with its $b$-ary expansion $\bold{a}=a_1a_2a_3\cdots$, if there exist a finite set $S=\{s_1, s_2, \cdots, s_l\}$ of primes and a strictly increasing sequence $\{n_i\}_{i=1}^\infty$ of positive integers such that for any $i$ and some positive number $0<\epsilon_{1.1}\leq 1$, in which $u_{n_i}$ is the numerator of the $n_i$-th ra

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof : Proof of Theorem \ref{['thm1']}
  • Corollary 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 49 more