Skolem–Mahler–Lech theorem


In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, an infinite arithmetic progression is full if there exist integers a and b such that the progression consists of all positive integers equal to b modulo a.
This result is named after Thoralf Skolem, Kurt Mahler, and Christer Lech. Its proofs use p-adic analysis.

Example

Consider the sequence
that alternates between zeros and the Fibonacci numbers.
This sequence can be generated by the linear recurrence relation
, starting from the base cases F = F = F = 0 and F = 1. For this sequence,
F = 0 if and only if i is either one or even. Thus, the positions at which the sequence is zero can be partitioned into a finite set and a full arithmetic progression.
In this example, only one arithmetic progression was needed, but other recurrence sequences may have zeros at positions forming multiple arithmetic progressions.

Related results

The Skolem problem is the problem of determining whether a given recurrence sequence has a zero. There exist an algorithm to test whether there are infinitely many zeros, and if so to find the decomposition of these zeros into periodic sets guaranteed to exist by the Skolem–Mahler–Lech theorem. However, it is unknown whether there exists an algorithm to determine whether a recurrence sequence has any non-periodic zeros.