Prove that $C(n,j)$ is periodic on $\mathbb{F}_2$?

Question

Fix j and consider the sequence of binomial coefficients $[C(n,j)]_n$ on $\mathbb{F}_2$. It seems like it is periodic, but I am not sure how to prove it, or to determine its period.

Answer 1

It is periodic indeed, since by Kummer's theorem, $\binom nj$ is even iff, for at least one of the binary digits of $j$ equal to $1$, the corresponding digit of $n$ is equal to $0$: $$2\mid\binom nj\iff\exists i,\quad j_i=1\text{ and }n_i=0.$$ The period is therefore the least power of $2$ which is greater than $j$.