Geodesic
On Positive Integer Solutions of the Equation xy + yz + xz = n
Canadian mathematical bulletin, Tome 39 (1996) no. 2, pp. 199-202

Voir la notice de l'article provenant de la source Cambridge University Press

As it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.
DOI : 10.4153/CMB-1996-024-5
Mots-clés : 11D09, 11R29, class numbers, diophantine equations, quadratic forms 
Hassan, Al-Zaid; Brindza, B.; Pintér, Á. On Positive Integer Solutions of the Equation xy + yz + xz = n. Canadian mathematical bulletin, Tome 39 (1996) no. 2, pp. 199-202. doi: 10.4153/CMB-1996-024-5
@article{10_4153_CMB_1996_024_5,
     author = {Hassan, Al-Zaid and Brindza, B. and Pint\'er, \'A.},
     title = {On {Positive} {Integer} {Solutions} of the {Equation} xy + yz + xz = n},
     journal = {Canadian mathematical bulletin},
     pages = {199--202},
     year = {1996},
     volume = {39},
     number = {2},
     doi = {10.4153/CMB-1996-024-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-024-5/}
}
TY  - JOUR
AU  - Hassan, Al-Zaid
AU  - Brindza, B.
AU  - Pintér, Á.
TI  - On Positive Integer Solutions of the Equation xy + yz + xz = n
JO  - Canadian mathematical bulletin
PY  - 1996
SP  - 199
EP  - 202
VL  - 39
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-024-5/
DO  - 10.4153/CMB-1996-024-5
ID  - 10_4153_CMB_1996_024_5
ER  - 
%0 Journal Article
%A Hassan, Al-Zaid
%A Brindza, B.
%A Pintér, Á.
%T On Positive Integer Solutions of the Equation xy + yz + xz = n
%J Canadian mathematical bulletin
%D 1996
%P 199-202
%V 39
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-024-5/
%R 10.4153/CMB-1996-024-5
%F 10_4153_CMB_1996_024_5

Kovâcs, K., About some positive solutions of the diophantine equation Σ1≤i<j≤n aiaj = m, Publ. Math. Debrecen, 40( 1992), 207–210. Google Scholar

Mordell, L. J., On the number of solutions in positive integers of the equation yz + zx + xy, Amer. J. Math. 45(1923), 1–4. Google Scholar

Mordell, L. J., Diophantine Equations, Academic Press, London 1969. Google Scholar

Oesterlè, J., Le problème de Gauss sur le nombre de classes, Enseign. Math. (2) 34(1988), 43—67. Google Scholar

Oesterlè, J., Nombres de classes des corps quadratiques imaginaires, Séminaire Nicolas Bourbaki, 1983— 1984, Exp. 631. Google Scholar

[S] Siegel, C. L., Abschzätzung von Einheiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. 2(1969), 71–86. Google Scholar

Taruzawa, T., On a theorem of Siegel, Japan. J. Math. 21(1951), 163–178. Google Scholar

Cité par Sources :