Originally Posted by

**Shanks** If we apply the euler function to the product of $\displaystyle q_i$ , and combined with the second equality, we have:

$\displaystyle 2^k-1=p^{\alpha-1}$ (&)

thus $\displaystyle \alpha$ is not greater than k.

combined with the first equality, gives p is less than the greatest prime among $\displaystyle q_i$.

from (&), we see that

$\displaystyle \prod_{i=1}^{k}q_i = p^\alpha+2 = p(2^k-1)+2 < 2^kp$

that is impossible since $\displaystyle q_i,p$ are distinct odd prime!

is it correct?