Geodesic
Théorie des nombres
Euclid meets Popeye: The Euclidean Algorithm for 2×2 Matrices
Bacher, Roland1
1 Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France
Comptes Rendus. Mathématique, Tome 361 (2023) no. G5, pp. 889-895

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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.

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DOI : 10.5802/crmath.451
Classification : 11A05, 11H06, 11J70

Bacher, Roland 1

1 Univ. Grenoble Alpes, Institut Fourier, 38000 Grenoble, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Euclid meets {Popeye:} {The} {Euclidean} {Algorithm} for $2\times 2$ {Matrices}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {889--895},
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Bacher, Roland. Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices. Comptes Rendus. Mathématique, Tome 361 (2023) no. G5, pp. 889-895. doi: 10.5802/crmath.451

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