HEXAGONbeta

A unimodular continued fraction for cubic irrationals

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We construct a single algorithm that assigns to every real number a recoverable sequence of positive integers which is eventually periodic exactly when the input is a cubic irrational. Its matrix digits are positive unimodular transformations on the two-dimensional simplex. Their products determine nested rational triangles whose diameters decrease by at least a factor of two at each step. An effectively specified partition combines Euclidean reduction with the dominant eigenvectors of cubic units. In both cubic signatures the matrix expansion eventually repeats a single matrix, whose Perron eigenvalue is a unit generating the input field.

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References

  1. C. Hermite, Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. (Continuation). , J. Reine Angew. Math. 40 (1850), 279--315. doi:10.1515/crll.1850.40.279 (https://doi.org/10.1515/crll.1850.40.279).DOI
  2. O. Karpenkov, On a periodic Jacobi--Perron type algorithm , Monatsh. Math. 205 (2024), 531--601. doi:10.1007/s00605-024-02006-5 (https://doi.org/10.1007/s00605-024-02006-5).DOI
  3. H. Řada, Š. Starosta, and V. Kala, Periodicity of general multidimensional continued fractions using repetend matrix form , Expo. Math. 42 (2024), Article 125571. doi:10.1016/j.exmath.2024.125571 (https://doi.org/10.1016/j.exmath.2024.125571).DOI
  4. L. Tagnon, A deterministic sin⁡2\sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates , preprint (2026). arXiv:2608.23281 (https://arxiv.org/abs/2608.23281).arXiv

Version history

  1. v1 · 6 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 12, 2026