On the Geometry of the Multiplicatively Closed Sets generated by at most Two Elements with arbitrarily Large Gaps, a constructive method

TL;DR

This paper explores the structure and gaps of multiplicatively closed sets generated by at most two elements, providing explicit constructions, geometric criteria, and proofs for arbitrarily large gaps, with applications to prime factorizations and multiplicative lines.

Contribution

It introduces a constructive method to find large gaps in such sets, establishes geometric criteria for their structure, and extends understanding of their properties using prime factorizations and space correspondence.

Findings

Existence of arbitrarily large gaps in multiplicatively closed sets generated by at most two elements.

A geometric correspondence between multiplicatively closed sets and points in a projective space.

Constructive proofs for large gaps without known prime factorizations at endpoints.

Abstract

We prove in Theorem $2.2$ that the multiplicatively closed subset generated by at most two elements in the set of natural numbers $\mathbb{N}$ has arbitrarily large gaps by explicitly constructing large integer intervals with known prime factorization for the end points, which do not contain any element from the multiplicatively closed set apart from the end points, which belong to the multiplicatively closed set. An Example $4.6$ is also illustrated. We also give a criterion in Theorems $7.8,7.12$ by using a geometric correspondence between maximal singly generated multiplicatively closed sets and points of the space $\mathbb{PF}^{\infty}_{\mathbb{Q}\geq 0}$ (refer to Theorem $7.5$) as to when a finitely generated multiplicatively closed set gives rise to a doubly multiplicatively closed line (refer to Definition $7.4$). We answer a similar Question $5.1$ partially about gaps in a multiply-generated multiplicative closed set, when it is contained in a doubly multiplicative closed set using Theorem $7.8$ and Theorem $7.17$. In the appendix Section $8$ we discuss another constructive proof (refer to Theorem $8.6$) for arbitrarily large gap intervals, where the prime factorization is not known for the right end-point unlike the constructive proof of the main result of the article in the case of multiplicatively closed set $\{p_1^ip_2^j\mid i,j\in \mathbb{N}\cup\{0\}\}$ with $\{p_1<p_2,Log_{p_1}(p_2)\}$ irrational for which the prime factorization is known for both the end-points of the gap interval via the stabilization sequence of the irrational $\frac {1}{Log_{p_1}(p_2)}$.