### Skolem's problem and prime powers

Ergodic Theory and Dynamical Systems Seminar

13th December 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

The terms in a linear recurrence sequence (LRS) satisfy a recurrence relation x_n = a_1x_{n-1} + a_2 x_{n-2} + \cdots + a_m x_{n-m}.

Skolem's problem studies the set of n\in\mathbb{N} such that x_n=0. A remarkable result of Skolem--Mahler--Lech states that, given a non-degenerate LRS, this set is a 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 x_n=0 when n is a prime power.

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