Let be a prime. .
the answer is and we don't need the condition to prove this let
by the Fermat's little theorem, in we have as a result which is Wilson's theorem, and
for all thus, for any so your sum, modulo ,
is equal to
Remark 1. this method gives another proof for the problem you asked in here.
Remark 2. there's a theorem (i don't remember its name) which says that for any prime number does anybody know the
name of this theorem? anyway, using this result we get this stronger result that, modulo your sum for is still equal to
ok, i found it! it's Wolstenholme's theorem.