Abstract
We prove that if the signed binomial coefficient
(−1
)
i
(
k
i
)
viewed modulo p is a periodic function of i with period h prime to p in the range
0≤i≤k
, then k+1 is a power of p, provided h is not too large compared to k. (In particular,
2h≤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∈G
for all
α∈G∖H
, then
G∪{0}
is a subfield.