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 $\alpha = \frac{\log(p_1)}{\log(p_2)}$irrational,wedescribe,inmainTheorem$\Omega$andinNote$1.5$, 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}}$usingprimaryandsecondaryconvergentsof$\alpha$. This suggests general Question $1.2$ for more than two generators which is still open.