Representation numbers of two octonary quadratic forms (Q2866991)

Let \(N(a_1,\ldots,a_4;n)\) denote the number of representations of an integer \(n\) by the form NEWLINE\[NEWLINEa_1(x_1^2 + x_1x_2 + x_2^2) + a_2(x_3^2+ x_3x_4 + x_4^2) + a_3(x_5^2 + x_5x_6 + x_6^2) + a_4(x_7^2 + x_7x_8 + x_8^2).NEWLINE\]NEWLINE In this paper the author derives formulae for \(N(1,1,1,2;n)\) and \(N(1,2,2,2;n)\) by using the method of Alaca, Alaca and Williams. These formulae are given in terms of \(\sigma_{3}(n)\). In the calculations he used the PARI software. The main theorem is as follows:NEWLINENEWLINETheorem. Let \(n\in\mathbb N\). Set \(n=2^a3^bN\), where \(a,b\in\mathbb N_0\), \(N\in\mathbb N\), \(\gcd(N,6)=1\) and \(N=\prod_{p\mid N}p^{a_p}\). Then NEWLINE\[NEWLINEN(1,1,1,2;n)=\frac 6{91}(2^{3a+4}+5)(3^{3b+2}+4)\prod_{p\mid N}\frac{p^{3a_p+3}-1}{p^3-1},\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEN(1,2,2,2;n)=\frac 6{91}(2^{3a+1}+5)(3^{3b+2}+4)\prod_{p\mid N}\frac{p^{3^{a_p}+3}-1}{p^3-1}.\tag{2}NEWLINE\]NEWLINE NEWLINENEWLINEThe needed evaluations of certain convolution sums are given in Section 3, and after the proof of the theorem in Section 4, the final section eight other octonary quadratic forms to which the method of this paper applies are listed.