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?