Date: Feb 23, 2013 1:56 PM
Author: Don Redmond
Subject: Congruence involving binomial coefficients

A colleague asked me if the following was true and I can't seem to give him an answer. Any help would be appreciated. He has a fair amount of numerical evidence, but just be a victim of the law of small numbers.

Let q be an odd prime such that n = 2q + 1 is also a prime. Let g be an even integer in the interval [2, q + 1]. Finally, let

F(n, g) = [C(n, g) - nC(q, g/2)]/2n,

where C(k, j) is the usual binomial coefficient k!/[j!(k-j)!]. The conjecture is

F(n, g) = 1 (mod q) if g = q + 1 and = 0 (mod q) if not.

Thanks in advance,