Skolem problem

Start a sequence with 2 and 1. Each new number is twice the latest number minus the previous one, giving 2, 1, 0, −1, and so on. Here the third term is zero. The Skolem problem asks for a method that always decides whether zero ever appears in any integer sequence made from fixed initial values and a fixed addition-and-subtraction rule of this kind. The rule may use any fixed number of preceding terms.[1] [2]

Given a fixed integer rule for producing a sequence, can an algorithm always decide whether zero ever occurs? The Skolem problem permits each new value to combine a fixed window of earlier values with fixed integer weights. If the window has length $k$, write $u_{n+k}=a_1u_{n+k-1}+\cdots+a_ku_n$, with integer coefficients and integer initial terms. This is an integer linear recurrence of order $k$; the target is whether some $n\ge0$ has $u_n=0$, allowing arbitrary $k$. Recording the window as a vector turns one step into multiplication by a fixed integer matrix. Zero occurrence is then whether that trajectory ever meets the set where a designated coordinate is zero—a hyperplane.[1] [2]

Computing terms one by one finds a zero if one occurs. If none occurs, that search runs forever without giving an answer. Some kinds of sequence can be handled, but no unconditional method is known that always stops with the correct yes or no answer for every allowed rule.[1] [2]

The Skolem–Mahler–Lech theorem says the set of zero indices is a finite union of arithmetic progressions and a finite set. An arithmetic progression repeats with a fixed step; the difficult additional zeros lack a general computable search bound. Splitting periodic patterns into separate subsequences reduces the problem to nonzero sequences with finitely many zeros. The 2026 paper proposes a bound on their last zero under a strengthened Cramér-type assumption for its specially defined good primes. Its unconditional result makes possible large-zero indices sparse, but sparsity does not exclude every late zero or decide every input.[1] [2]

A repeated calculation can be programmed to stop when one of its stored numbers reaches zero. The sequence question then becomes a precise question about whether that loop ever stops. These simple rules give a useful testing ground for automated verification: proving a property for infinitely many steps requires more than watching a long run.[2]

The matrix viewpoint connects the problem to linear loops and reachability, but also reveals a sharp mathematical distinction: reaching one specified state is decidable, whereas reaching an entire hyperplane is the open Skolem target. A trajectory can cross that hyperplane through exact cancellation between growing or oscillating terms. Understanding when such cancellations can be ruled out links arithmetic structure to finite certificates about an infinite computation, giving both number theory and verification a shared concrete problem.[1] [2]