On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia¹

¹ Department of Mathematics Jiangsu University Zhenjiang, Jiangsu 212013, P.R. China

Colloquium Mathematicum, Tome 135 (2014) no. 1, pp. 139-145.

Résumé

For natural numbers $a, b$ and positive integer $n$, let $R(a,b;n)$ denote the number of representations of $n$ in the form $$ \sum _{i=1}^a (x_i^2+x_iy_i+y_i^2)+2\sum _{j=1}^b(u_j^2+u_jv_j+v_j^2). $$ Lomadze discovered a formula for $R(6,0;n)$. Explicit formulas for $R(1,5;n)$, $R(2,4;n)$, $R(3,3;n)$, $R(4,2;n)$ and $R(5,1;n)$ are determined in this paper by using the $(p; k)$-parametrization of theta functions due to Alaca, Alaca and Williams.

Détail

DOI : 10.4064/cm135-1-11

Keywords: natural numbers positive integer n denote number representations form sum sum lomadze discovered formula explicit formulas determined paper using parametrization theta functions due alaca alaca williams