Let K be a field of field characteristic 0 (e.g., the rationals Q) and let {u_n} be a sequence of elements of K which satisfies a difference equation of the form 0 = c_0 u_n + c_1 u_(n + 1) + ... + c_k u_(n + k), where the coefficients c_i are fixed elements of K. Then, for any c element K, we have either u_n = c for only finitely many values of n, or u_n = c for the values of n in some arithmetic progression. The proof involves embedding certain fields inside the p-adic numbers Q_p for some prime p, and using properties of zeros of power series over Q_p (Strassman's theorem).

arithmetic progression | p-adic number | Strassman's theorem