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

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 $S=\{p_1^i{p}_2^j\mid i,j\in \mathbb{N}\cup \{0\}\}$ using primary and secondary convergents of $\alpha$. This suggests general Question $1.2$ for more than two generators which is still open.

15 pages, To Appear in Proceedings Mathematical Sciences, Indian Academy of Sciences, Accepted in January 2020

References in corpus (1)

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