We define recursively $p_1=2,p_2=3$ and $$p_{n}= \min_{F_{n-1}}|A-B| $$ Where $F_{n-1}=((A,B) |\gcd (A,B)=1,A,B $ are products of powers of $p_i$s $\\ \mathrm{and} \\ ( p_i |A \mathrm{\\ or\}\\ B) \\ 1 \leq i \leq n-1 , |A-B| \not =1 )$

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