On the Factorization of Two Adjacent Numbers in Multiplicatively Closed Sets Generated by Two Elements

TL;DR

This paper characterizes the factorization of two consecutive numbers in a multiplicatively closed set generated by two numbers with irrational log ratio, using continued fraction convergents, and discusses open questions for more generators.

Contribution

It provides a detailed description of the factorization of adjacent numbers in such sets using continued fractions, extending understanding of their structure.

Findings

Factorization of adjacent numbers is described using primary and secondary convergents.

The work highlights the role of irrational log ratios in the structure of multiplicatively closed sets.

Open questions are posed for sets with more than two generators.

Abstract

For two natural numbers $1<p_1<p_2$, with $\alpha = \frac{\log(p_1)}{\log(p_2)}$ irrational, we describe, in main Theorem $\Omega$ and in Note $1.5$, the factorization of two adjacent numbers in the multiplicatively closed subset $S = \{p_1^ip_2^j\mid i,j\in \mathbb{N}\cup\{0\}\}$ using primary and secondary convergents of $\alpha$. This suggests general Question $1.2$ for more than two generators which is still open.