Modular periodicity of binomial coefficients
Abstract
Description
We prove that if the signed binomial coefficient $(-1)^i\binom{k}{i}$ viewed modulo p is a periodic function of i with period h prime to p in the range $0\le i\le k$, then k+1 is a power of p, provided h is not too large compared to k. (In particular, $2h\le k$ suffices.) As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that $1-α\in G$ for all $α\in G\setminus H$, then $G\cup\{0\}$ is a subfield.
8 pages. Somehow, the references were missing in the previous version. An error in the abstract (but not in the main text) of the printed version is corrected here: h needs to be prime to p
8 pages. Somehow, the references were missing in the previous version. An error in the abstract (but not in the main text) of the printed version is corrected here: h needs to be prime to p