Hartmanis–Stearns conjecture

Determine whether there exists a real algebraic irrational number whose binary, or more generally base-, expansion can be computed in linear time by a Turing machine.

Background

The paper introduces the computational-complexity question of whether algebraic irrational numbers can have efficiently computable digit expansions. Hartmanis and Stearns asked whether a real algebraic irrational number can have a binary or integer-base expansion computable in linear time by a Turing machine. The weaker finite-automaton version, known as Cobham's conjecture, is described as having been solved, whereas the original Hartmanis–Stearns conjecture remains unresolved.

References

While the Hartmanis--Stearns conjecture remains open, Cobham's conjecture was proved by Adamczewski, Bugeaud and Luca, based on Schlickewei's $\mathfrak{p}$-adic subspace theorem.

— Transcendence of dendric beta-expansions with pure discrete spectrum  (2610.00856 - Seo et al., 1 Oct 2026) in Section 1, Introduction and main result