Démonstration de la conjecture de Dumont

Bodo Lass

doi:10.1016/j.crma.2005.10.018

Théorie des nombres/Combinatoire

Démonstration de la conjecture de Dumont
[A proof of Dumont's conjecture]

Presented by: Étienne Ghys

Bodo Lass ¹

¹Institut Camille Jordan, UMR 5208 du CNRS, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 713-718

Résumé (Original language)
Abstract

Soit

r_{k}^{1 (2)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1}^{2} + x_{2}^{2} + \dots + x_{k}^{2}, x_{i} \equiv 1 (\mod 2), 1 ⩽ i ⩽ k} |,

c_{k}^{1 (4)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1} x_{2} + x_{2} x_{3} + \dots + x_{k - 1} x_{k} + x_{k} x_{1}, x_{i} \equiv 1 (4)} |,

c_{k}^{3 (4)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1} x_{2} + x_{2} x_{3} + \dots + x_{k - 1} x_{k} + x_{k} x_{1}, x_{i} \equiv 3 (4)} | .

Dumont a conjecturé l'identité

r_{k}^{1 (2)} (n) = c_{k}^{1 (4)} (n) - {(- 1)}^{k} c_{k}^{3 (4)} (n)

qui généralise, notamment, les résultats classiques de Lagrange, Gauß, Jacobi et Kronecker sur les décompositions de tout entier en deux, trois et quatre carrés. Nous donnons une preuve combinatoire de la conjecture de Dumont.

Let

r_{k}^{1 (2)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1}^{2} + x_{2}^{2} + \dots + x_{k}^{2}, x_{i} \equiv 1 (\mod 2), 1 ⩽ i ⩽ k} |,

c_{k}^{1 (4)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1} x_{2} + x_{2} x_{3} + \dots + x_{k - 1} x_{k} + x_{k} x_{1}, x_{i} \equiv 1 (4)} |,

c_{k}^{3 (4)} (n) : = | {(x_{1}, x_{2}, \dots, x_{k}) \in N^{k} | n = x_{1} x_{2} + x_{2} x_{3} + \dots + x_{k - 1} x_{k} + x_{k} x_{1}, x_{i} \equiv 3 (4)} | .

Dumont has conjectured the identity

r_{k}^{1 (2)} (n) = c_{k}^{1 (4)} (n) - {(- 1)}^{k} c_{k}^{3 (4)} (n)

, which generalizes, in particular, the classical results of Lagrange, Gauß, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont's conjecture.

Received: 2005-07-11
Accepted: 2005-10-11
Published online: 2005-11-14

DOI: 10.1016/j.crma.2005.10.018

Author's affiliations:

Bodo Lass ¹

¹ Institut Camille Jordan, UMR 5208 du CNRS, Université Claude Bernard Lyon 1, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

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     author = {Bodo Lass},
     title = {D\'emonstration de la conjecture de {Dumont}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {713--718},
     year = {2005},
     publisher = {Elsevier},
     volume = {341},
     number = {12},
     doi = {10.1016/j.crma.2005.10.018},
     language = {fr},
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Bodo Lass. Démonstration de la conjecture de Dumont. Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 713-718. doi: 10.1016/j.crma.2005.10.018

References
Cited by

[1] G. Andrews The Theory of Partitions, Teoriya razbienii, Cambridge University Press, 1998 (en russe)

[2] D. Dumont, A conjecture on sums of any number of odd squares, Prépublication

[3] A. Weil Sur les sommes de trois et quatre carrés, Enseign. Math. II. Sér., Volume 20 (1974), pp. 215-222

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