  A function on a (generally infinite) graph $\G$ with values in a field $K$ of
characteristic 2 will be called {\it harmonic} if its value at every vertex of
$\G$ is the sum of its values over all adjacent vertices. We consider binary
pluri-periodic harmonic functions $f: \Z^s\to\F_2=\GF(2)$ on integer lattices,
and address the problem of describing the set of possible multi-periods $\bar
n=(n_1,...,n_s)\in\N^s$ of such functions. Actually this problem arises in the
theory of cellular automata. It occurs to be equivalent to determining, for a
certain affine algebraic hypersurface $V_s$ in $\A_{\bar\F_2}^s$, the torsion
multi-orders of the points on $V_s$ in the multiplicative group
$(\bar\F_2^\times)^s$. In particular $V_2$ is an elliptic cubic curve. In this
special case we provide a more thorough treatment. A major part of the paper is
devoted to a survey of the subject.
