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The monoid of numbers of the form 1 < a^q /b^p < a

Laurent Fallot

TL;DR

This work generalizes the Collatz-context inquiry to rational numbers of the form $\frac{a^q}{b^p}$ with coprime $1<a<b$, defining $\mathcal{F}_{(a,b)}$ as those with $1 \le \frac{a^q}{b^p} < a$. A monoid structure is established via the isomorphism $\phi_{(a,b)}: (\mathbb{N},+ ,0) \to (\mathcal{F}_{(a,b)},\star,1)$, enabling a precise enumeration of elements and a robust, globally defined notion of both minimum and maximum record holders. Two interdependent sequences $u_{(a,b)}$ and $v_{(a,b)}$ of record holders are constructed and analyzed; they converge to $1$ and $a$, respectively, and together imply that $\mathcal{F}_{(a,b)}$ is dense in $[1,a]_{\mathbb{R}}$. The results extend to the density of scaled copies $a^k\mathcal{F}_{(a,b)}$, yielding a dense cover of $\mathbb{R}^+$ and providing a framework linking monoid structure to real-analytic density phenomena in a number-theoretic setting.

Abstract

This paper is a study of the set of rational numbers of the form 1 < a^q /b^p < a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.

The monoid of numbers of the form 1 < a^q /b^p < a

TL;DR

This work generalizes the Collatz-context inquiry to rational numbers of the form with coprime , defining as those with . A monoid structure is established via the isomorphism , enabling a precise enumeration of elements and a robust, globally defined notion of both minimum and maximum record holders. Two interdependent sequences and of record holders are constructed and analyzed; they converge to and , respectively, and together imply that is dense in . The results extend to the density of scaled copies , yielding a dense cover of and providing a framework linking monoid structure to real-analytic density phenomena in a number-theoretic setting.

Abstract

This paper is a study of the set of rational numbers of the form 1 < a^q /b^p < a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.
Paper Structure (23 sections, 23 theorems, 33 equations, 1 table)

This paper contains 23 sections, 23 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Conventions and quick properties
  4. The function ${\phi}_{(a,b)}$ is an isomorphism of monoids.
  5. Record holders
  6. Base case
  7. Induction case
  8. Case 1: $1<{{u}_{(a,b)}}_{i} {{v}_{(a,b)}}_{i} < a$.
  9. Case 2: $a\leq {{u}_{(a,b)}}_{i} {{v}_{(a,b)}}_{i} < a^2$.
  10. Case 1: ${{u}_{(a,b)}}_{i} {{v}_{(a,b)}}_{i} < a$.
  11. Case 2: $a \leq {{u}_{(a,b)}}_{i} {{v}_{(a,b)}}_{i}$.
  12. Base cases
  13. Induction step
  14. Minimum record holders.
  15. Maximum record holders.
  16. ...and 8 more sections

Key Result

Proposition 3.1

Given two co-prime positive integers $1<a<b$, the only integer that ${\mathcal{F}}_{(a,b)}$ contains is $1$.

Theorems & Definitions (43)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Definition 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 33 more