On the representations of a positive integer by certain classes of quadratic forms in eight variables

B. Ramakrishnan; Brundaban Sahu; Anup Kumar Singh

paper

On the representations of a positive integer by certain classes of quadratic forms in eight variables

arXiv:1607.04764

Abstract

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_{1} x_{1}^{2} + a_{2} x_{2}^{2} + a_{3} x_{3}^{2} + a_{4} x_{4}^{2} + b_{1} (x_{5}^{2} + x_{5} x_{6} + x_{6}^{2}) + b_{2} (x_{7}^{2} + x_{7} x_{8} + x_{8}^{2})$ , where $a_{1} \leq a_{2} \leq a_{3} \leq a_{4}$ , $b_{1} \leq b_{2}$ and $a_{i}$ 's $\in {1, 2, 3}$ , $b_{i}$ 's $\in {1, 2, 4}$ . We also determine formulas for the number of representations of a positive integer by the quadratic forms $(x_{1}^{2} + x_{1} x_{2} + x_{2}^{2}) + c_{1} (x_{3}^{2} + x_{3} x_{4} + x_{4}^{2}) + c_{2} (x_{5}^{2} + x_{5} x_{6} + x_{6}^{2}) + c_{3} (x_{7}^{2} + x_{7} x_{8} + x_{8}^{2})$ , where $c_{1}, c_{2}, c_{3} \in {1, 2, 4, 8}$ , $c_{1} \leq c_{2} \leq c_{3}$ .

18 pages, 7 tables. arXiv admin note: substantial text overlap with arXiv:1607.03809