Comptes Rendus
Number theory
Euclid meets Popeye: The Euclidean Algorithm for 2×2 Matrices
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 889-895.

An analogue of the Euclidean algorithm for square matrices of size 2 with integral non-negative entries and positive determinant n defines a finite set (n) of Euclid-reduced matrices corresponding to elements of {(a,b,c,d) 4 |n=ab-cd,0c,d<a,b}. With Popeye’s help (acknowledged by his appearance in the title; he refused co-authorship on the flimsy pretext of a weak contribution due to a poor spinach-harvest) on the use of sails of lattices we show that (n) contains d|n,d 2 n d+1-n d elements.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/crmath.451
Classification: 11A05, 11H06, 11J70

Roland Bacher 1

1 Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Roland Bacher. Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 889-895. doi : 10.5802/crmath.451. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.451/

[1] Vladimir I. Arnolʼd Higher dimensional continued fractions, Regul. Chaotic Dyn., Volume 3 (1998) no. 3, pp. 10-17 | Zbl

[2] Roland Bacher A Quixotic Proof of Fermat’s Two Squares Theorem for Prime Numbers (to appear in Am. Math. Mon.)

[3] Bruno Gruber Alternative formulae for the number of sublattices, Acta Crystallogr., Sect. A, Volume 53 (1997) no. 6, pp. 807-808 | DOI | Zbl

[4] N. J. A. Sloane The On-Line Encyclopedia of Integer Sequences, 2010 (http://oeis.org)

[5] Yi Ming Zou Gaussian binomials and the number of sublattices, Acta Crystallogr., Sect. A, Volume 62 (2006) no. 5, pp. 409-410 | DOI | Zbl

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