You seem to be describing P(x) = ((x+1)

^{p}-x

^{p}-1)/p

(only you have "+1/p" still dangling at the end. I will subtract it for symmetry).

After binomial expansion:

It is obvious that for p>=7, the middle terms will have a factor

*precprime(p)*; it follows from

,

where k = (p-1)/2. This is because

*precprime(p)* will be present in the denominator, but not in the numerator.

Note that your own observation that "the two most inner terms are divisible by Q(p-1)" is not holding for p<7 (e.g. for p=5, "the two most inner terms" are

**not** divisible by 3), and the reason for this is obvious. For p>=7, it will hold, - as indeed shown above.