A note on Buell’s Theorem on length four Büchi sequences

Fabrice Jaillet; Xavier Vidaux

doi:10.56754/0719-0646.2701.001

Authors

Fabrice Jaillet UCBL, CNRS, INSA Lyon, LIRIS, UMR5205, F-69622 Villeurbanne, France. https://orcid.org/0000-0002-7330-8116
Xavier Vidaux Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática, Casilla 160 C, Chile. https://orcid.org/0000-0001-7680-031X

DOI:

https://doi.org/10.56754/0719-0646.2701.001

Keywords:

Representation of systems of quadratic forms, Büchi’s n-squares problem, second difference of squares

Abstract

Büchi sequences are sequences whose second difference of squares is the sequence (2,..., 2), like for instance (6, 23, 32, 39) – so they can be seen as a generalization of arithmetic progressions. No (non-trivial) length 5 Büchi sequence is known to exist. Length four Büchi sequences were parameterized by D. A. Buell in 1987. We revisit his theorem, fixing the statement (about 26% of the Büchi sequences from R. G. E. Pinch's 1993 table were missed), and giving a much simpler proof.

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References

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