Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i1$ and $q'^i1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, $\{r_6\}=\{u_6\}$ and $\{r_4,p\}=\{u_4,p'\}$. Now I want to know if it is possible that $p\neq p'$?
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2$\begingroup$ Primitive with respect to q and i or wrt p and alphai? And if there is more than one candidate for r_i, which do you pick? $\endgroup$ – The Masked Avenger Aug 24 '13 at 4:54

$\begingroup$ Seems very close to mathoverflow.net/questions/91420/primitiveprimedivisor $\endgroup$ – minar Aug 24 '13 at 6:16