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

For two natural numbers $1<p_1<p_2$, with $α= \frac{\log(p_1)}{\log(p_2)}$ irrational, we describe, in main Theorem $Ω$ 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 $α$. This suggests general Question $1.2$ for more than two generators which is still open.

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