linear combinations of binomial coefficients

[Max A.]

On 8/2/06, Max A. <maxale at gmail.com> wrote:

> It was hypothesized that for any set of n>=2 distinct positive
> integers c1<c2<...<cn there exists a set integers k1,...,kn such that
> the following linear combination of binomial coefficients is divisible
> by p^(2n-1) for *all* large enough prime p:
> k1*C(c1*p,p) + k2*C(c2*p,p) + ... + kn*C(cn*p,p)

It was also conjectured that the "outer" coefficients k1,...,kn are
defined uniquely as soon as k1>0 and gcd(k1,...,kn)=1.

[...]

> The sequence of lower bounds for prime p seems to be just a sequence
> of all prime numbers with occasional "drops" to the absolute minimum
> value 3 (at n=5,8,11,13,14,...)

A. Adamchuk has suggested that sequence 5,8,11,13,14,... is A053726
meanning that the lower bound is 3 if and only if (2n-1) is not prime.
If (2n-1) is prime then the lower bound equals next prime after
(2n-1).

Max