Démonstration de la conjecture de Dumont

Letrk1(2)(n):=|{(x1,x2,…,xk)∈Nk|n=x12+x22+⋯+xk2,xi≡1(mod2),1⩽i⩽k}|,ck1(4)(n):=|{(x1,x2,…,xk)∈Nk|n=x1x2+x2x3+⋯+xk−1xk+xkx1,xi≡1(4)}|,ck3(4)(n):=|{(x1,x2,…,xk)∈Nk|n=x1x2+x2x3+⋯+xk−1xk+xkx1,xi≡3(4)}|. Dumont has conjectured the identity rk1(2)(n)=ck1(4)(n)−(−1)kck3(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. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005).