On the gaps in multiplicatively closed sets generated by at most two elements

We prove in the the main theorem, Theorem 3.2, that the multiplicatively closed subset of natural numbers, generated by two elements 1<p1<p2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p_1<p_2$$\end{document} with α=logp1logp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\frac{\log \ p_1}{\log \ p_2}$$\end{document} irrational, has arbitrarily large gaps by explicitly constructing large integer intervals, with known factorization for the endpoints in terms of generators p1,p2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1,p_2$$\end{document} obtained from the stabilization sequence of the irrational α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} (Definition 3.1). Example 5.6 is also illustrated. In the Appendix, for a finitely generated multiplicatively closed subset of natural numbers, we mention another constructive proof (refer to Theorem A.1) for arbitrarily large gap intervals, where the factorization of the right endpoint is not known in terms of generators unlike in the constructive proof of the main result. The suggested general Question 1.1 remains still open.