Lehmer's problem over a finite field

Does there exist an odd prime $p>3$, and integers $m>2$ and $n_k>0$ such that

$$p^{\sum_{k=1}^m n_k} \equiv 1 \bmod \prod_{k=1}^m (p^{n_k}-1) ?$$

where if some n_i are the same , the same number appears at most r(n,p) times. r(n,p) is the number of irreducible polynomials of degree n over the finite field F_p.

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m is larger than 1 –  mason Apr 15 '12 at 15:53
Yes, for example $p=3$. See here: mathoverflow.net/questions/81816/… –  Ralph Apr 15 '12 at 16:22
It may be helpful, if you could provide more background about your question. In particular, why do you need p>3 and what's the reason for the condition involving r(n,p) ? –  Ralph Apr 16 '12 at 0:02
the decomposition of a polynomial –  mason Apr 16 '12 at 14:53