Hermite’s periodicity problem for cubic irrationals; Karpenkov’s Problem 4

Hermite posed the problem in 1848: find an algorithmic representation that is eventually periodic exactly for cubic irrationals. Karpenkov’s Problem 4 concerns this question for complex cubic irrationals, where the required universal periodicity theorem is not known.

Known results

• Hermite, 1848: formulated the periodic-representation problem for cubic irrationals.
• A 2015 ternary continued-fraction construction gives periodic representations for all cubic irrationals, but requires the minimal polynomial and is not an algorithm defined on all real inputs.
• Karpenkov’s $\sin^2$-algorithm proves periodicity for all totally real cubic irrationalities; the complex-conjugate-root case remains open.
• Other constructions cover restricted families or special cubic fields, not every complex cubic irrational.

August 24, 2026 finite-region certification

A preprint, “A deterministic $\sin^2$-type algorithm for complex cubic irrationalities with exact periodicity certificates,” reports a deterministic extension with tie-breaking and exact certificates for substantial finite computational domains. This is claimed computational progress, not a proof of eventual periodicity for every complex cubic irrational, and has not been independently verified here.

Current status (as of August 2026): Periodicity is proved for the totally real cubic case and claimed for certified finite complex domains, while the universal complex case remains open.