Subsets of $\mathbb N$ with a finite number of prime factors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:38:00Z http://mathoverflow.net/feeds/question/61124 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61124/subsets-of-mathbb-n-with-a-finite-number-of-prime-factors Subsets of $\mathbb N$ with a finite number of prime factors Jens Reinhold 2011-04-09T09:58:41Z 2011-04-12T18:39:23Z <p>We call a subset $A = \{a_1, a_2, a_3, \dots\}$ of $\mathbb N$ with $a_1 &lt; a_2 &lt; \dots$ 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.</p> http://mathoverflow.net/questions/61124/subsets-of-mathbb-n-with-a-finite-number-of-prime-factors/61133#61133 Answer by Gjergji Zaimi for Subsets of $\mathbb N$ with a finite number of prime factors Gjergji Zaimi 2011-04-09T11:18:52Z 2011-04-09T11:18:52Z <p>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 <a href="http://en.wikipedia.org/wiki/S-unit" rel="nofollow">S-unit</a> theorem.</p>