Comptes Rendus
Théorie des nombres
Euclid meets Popeye: The Euclidean Algorithm for 2×2 Matrices
Roland Bacher1
1 Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France
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 ℛ(n) of Euclid-reduced matrices corresponding to elements of {(a,b,c,d)∈ℕ 4 |n=ab-cd,0≤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 ∑ d|n,d 2 ≥n d+1-n d elements.

Reçu le :
Accepté le :
Accepté après révision le :
Publié le :
DOI : 10.5802/crmath.451
Classification : 11A05, 11H06, 11J70
Roland Bacher 1

1 Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G5_889_0,
     author = {Roland Bacher},
     title = {Euclid meets {Popeye:} {The} {Euclidean} {Algorithm} for $2\times 2$ {Matrices}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {889--895},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.451},
     language = {en},
}
TY  - JOUR
AU  - Roland Bacher
TI  - Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 889
EP  - 895
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.451
LA  - en
ID  - CRMATH_2023__361_G5_889_0
ER  - 
%0 Journal Article
%A Roland Bacher
%T Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices
%J Comptes Rendus. Mathématique
%D 2023
%P 889-895
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.451
%G en
%F CRMATH_2023__361_G5_889_0
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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Three examples of three-dimensional continued fractions in the sense of Klein

Oleg Karpenkov

C. R. Math (2006)

A Riemannian approach to strain measures in nonlinear elasticity

Patrizio Neff; Bernhard Eidel; Frank Osterbrink; ...

C. R. Méca (2014)

Hunting for Cold Exoplanets via Microlensing

Jean-Philippe Beaulieu

C. R. Phys (2023)