I try not to consider the fact , i just regard it as the coefficient of a term in

We know is the coefficient of in the expansion of while is the coefficient of in the expansion of . Therefore , is the coefficient of of .

I would like to show that all the terms that the degrees are the multiple of prime are congruent to zero modulo .

I consider

Note that all the terms in are divisible by .

Then i consider the second polynomial , the annoying one .

I use the fact to simplify :

Then we are done , haha , let's get back the first polynomial whose degree ranges from to so they are prime to . Consider , we find that the degrees are all mutiple of so they become useless in the expansion if we are evaluating the coefficient of since no terms in could make the 'connection' to them . Therefore , the coefficient we are considering ( of ) is necessarily the mutiple of , one from and one from .

Remarks :

I don't like this result ... i am going to show that

For a prime we always have

:

Let's check my solution , write or

Then the coefficient of in is congruent to ( ) that in . It suffices to show that the coefficient of in is congruent to zero modulo .

ie :

.

Please see this post , http://www.mathhelpforum.com/math-he...nt-149645.html

It is not a help begging but a challenge , let's finish the proof by solving my challenge .