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

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 $\frac {1}{Log_{p_1}(p_2)}$.