Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices

Roland Bacher

doi:10.5802/crmath.451

Number theory

Euclid meets Popeye: The Euclidean Algorithm for

2 \times 2

Matrices

Roland Bacher ¹

¹ Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 889-895

Abstract

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 $ℛ (n)$ of Euclid-reduced matrices corresponding to elements of ${(a, b, c, d) \in ℕ^{4} | n = a b - c d, 0 \leq c, d < a, b}$ . 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 $ℛ (n)$ contains $\sum_{d | n, d^{2} \geq n} (d + 1 - \frac{n}{d})$ elements.

Received: 2022-10-29
Accepted: 2022-11-29
Revised after acceptance: 2022-12-07
Published online: 2023-07-18

DOI: 10.5802/crmath.451

Classification: 11A05, 11H06, 11J70

Author's affiliations:

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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     author = {Roland Bacher},
     title = {Euclid meets {Popeye:} {The} {Euclidean} {Algorithm} for $2\times 2$ {Matrices}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {889--895},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     doi = {10.5802/crmath.451},
     language = {en},
}

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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

References
Cited by

[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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