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Comptes Rendus Mathématique
Volume 341, n° 12
pages 713-718 (décembre 2005)
Doi : 10.1016/j.crma.2005.10.018
Received : 11 July 2005 ;  accepted : 11 October 2005
Démonstration de la conjecture de Dumont
A proof of Dumontʼs conjecture
 

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

Résumé

Soit
rk1(2)(n):=|{(x1,x2,,xk)Nk|n=x12+x22++xk2,xi1(mod2),1ik}|,ck1(4)(n):=|{(x1,x2,,xk)Nk|n=x1x2+x2x3++xk-1xk+xkx1,xi1(4)}|,ck3(4)(n):=|{(x1,x2,,xk)Nk|n=x1x2+x2x3++xk-1xk+xkx1,xi3(4)}|. Dumont a conjecturé lʼidentité   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. Pour citer cet article : B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.
Abstract

Let
rk1(2)(n):=|{(x1,x2,,xk)Nk|n=x12+x22++xk2,xi1(mod2),1ik}|,ck1(4)(n):=|{(x1,x2,,xk)Nk|n=x1x2+x2x3++xk-1xk+xkx1,xi1(4)}|,ck3(4)(n):=|{(x1,x2,,xk)Nk|n=x1x2+x2x3++xk-1xk+xkx1,xi3(4)}|. Dumont has conjectured the identity  , 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 Dumontʼs conjecture. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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