We define recursively 
$p_1=2,p_2=3$
and 
$$p_{n}= \min_{(A,B)\in F_{n-1}}|A-B| $$
Where
$$
\begin{split}
F_n=\{(A,B) |&\gcd (A,B)=1,\quad |A-B| \not =1,
\\\
&\text{both $A$ and $B$ are products of powers of $p_i$ for $i\le n$},
\\\
&\text{for each $i\le n$, either $p_i |A$ or $p_i |B$}\}
\end{split}
$$

Is always $p_n$ the $n-th$ prime?