OK when I saw this two methods immediately came to mind. One was finding some bijective function such that is a root of the polynomial (from now on call it ) only if is not a root. In this case, (by Occam's razor) the function would most likely be something like However, this didn't work which brings me to method two - finding lots of double roots. Now the best way to do this is to take derivatives (which is further motivated by the structure of itself). So let's do that!


From now on we work in Some important things to keep in my mind will be the simple generalization of Wilson's Theorem given by and Fermat's Little Theorem. Now note that for any we have

Noting that the maps and are bijective in we have that

Now consider the polynomial in We have for any that which implies that every root of is a double root which implies the same for Hence we have the desired result.

Hmm this is pretty hard - I can rewrite it in terms of generating functions but I don't know if it helps