We call a subset $A = \{a\_1, a\_2, a\_3, \dots\}$ of $\mathbb N$ with $a\_1 < a\_2 < \dots $ transparent if $a\_{k+1} - a\_k$ goes to $\infty$ as $k \rightarrow \infty$. Is the following true? For every finite set $P$ of prime numbers, the set $A\_P := \{p\_1 \cdots p\_s : p\_i \in P\}$ is transparent.



Subsets of $\mathbb N$ with a finite number of prime factors
Proving that $a\_{k+1}-a\_k\to \infty$ reduces to proving that $$a\_{k+1}-a\_k=n$$ has finitely many solutions for every $n$. If you let $S$ be the set of prime divisors of $n$ union $P$, then this follows from the [S-unit](http://en.wikipedia.org/wiki/S-unit) theorem.


