Skolem's Problem and Prime Powers

In this talk we are interested in the following Skolem's problem. Given a linear recurrence sequence (u_n), i.e. both the recurrence relation and initial conditions, determine whether or not there exists an n such that u_n=0. A remarkable result of Skolem--Mahler--Lech states that, if (u_n) is non-degenerate, the set of zeroes is the union of a finite set together with a finite number of (infinite) arithmetic progressions. However, the proof is non-constructive and the decidability of Skolem's problem remains open---a situation described as ‘faintly outrageous’ by Tao and a ‘mathematical embarrassment’ by Lipton.

We shall review some recent results in this field and indicate ongoing work determining whether u_n=0 when n is a prime power.