More precise conjecture for linear combinations of binomial coefficients

[Max A.]

Roland,

I've just found that this result is precisely formulated in proved at
http://www.dms.umontreal.ca/~andrew/Binomial/highpower.html

Max

On 8/4/06, Roland Bacher <Roland.Bacher at ujf-grenoble.fr> wrote:
>
> Here is a more precise (and slightly more general)
> version of a nice conjecture presented yesterday by Max:
>
> Let c_1,...,c_n \in Z be n integers.
> Let k_1,..k_n \in Z be a solution of the linear system given by
>
> (*)    sum_{j=1}^n k_j {c_j x \choose x}= 0
>
> for x=1,3,5,..,2n-3 (where {c_j x \choose x} is a binomial
> coefficient).
>
> Conjecture: We have have then
>
> sum_{j=1}^n k_j {c_j p \choose p}= 0 modulo p^{2n-1}
>
> for all except finitely many primes p.
>
> Remarks: - Choosing c_j=0 or an equality c_j=c_i is not interesting.
>
> - For a generic choice of c_1,...,c_n, the linear system (*)
> has a unique solution (k_1,..,k_n)\in Z^n, up to a constant.
>
> - Multiplying a given solution (k_1,..,k_n) by a suitable integer,
> the identity of the conjecture (if true) can be assumed to hold for all
> primes. Working with valuations, the identity makes of course sense
> for rational solutions (k_1,..k_n)\in Q^n of (*).
>
> Roland Bacher