Skolem-Mahler-Lech Theorem
If {a_0, a_1, ...} is a recursive sequence, then the set of all k such that a_k = 0 is the union of a finite (possibly empty) set and a finite number (possibly zero) of full arithmetical progressions, where a full arithmetic progression is a set of the form {r, r + d, r + 2d, ...} with r element [0, d).