Representation Numbers of Two Octonary Quadratic Forms

Abstract

Let N(a1, …, a4; n) denote the number of representations of an integer n by the form [Formula: see text]. In this paper we derive formulae for N(1, 1, 1, 2; n) and N(1, 2, 2, 2; n). These formulae are given in terms of σ3(n).

Representation numbers by sums of quadratic forms $x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}$ in sixteen variables

“…To prove the theorem we need the representation number formulae for the following octonary quadratic forms, is given by Lomadze [6]. Köklüce [3] has obtained formulae for the number of representation of a positive integer l by the forms f 2 and f 3 : He has proved that, Proof. (i) We just prove part (i) in detail as the rest can be proved similarly.…”