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

Volume 135 / 2014

Ernest X. W. Xia Colloquium Mathematicum 135 (2014), 139-145 MSC: Primary 11E25; Secondary 11E20, 11A25. DOI: 10.4064/cm135-1-11

Abstract

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.

Authors

  • Ernest X. W. XiaDepartment of Mathematics
    Jiangsu University
    Zhenjiang, Jiangsu 212013, P.R. China
    e-mail

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