Sum of powers factorization

Hi everybody,

I think that:

$\displaystyle

N=\sum_{i=0}^{s}p^i=p^s+...+1 \quad \quad s>1

$

Is factorized in prime numbers (according theorem of arithmetic), that are square free.

In other words its dinstict primes (if multiplied) give as result number N.

i.e.

$\displaystyle

N=p_1^{a_1}\cdot...\cdot p_k^{a_k}

$

where

$\displaystyle

{a_1}=...={a_k}=1

$

**Is this correct?**

I think this can be handled with Congruence of Clausen and von Staudt proof. Unfortunatelly i cannot find any link (with proof) into internet.

Thank you a lot