Representations by Certain Octonary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6

Abstract

We determine formulae for the numbers of representations of a positive integer by certain octonary quadratic forms whose coefficients are 1, 2, 3 and 6.

Representations by certain octonary quadratic forms with coefficients 1, 3 or 9

“…. , x 8 ) ∈ Z 8 | n = x 2 1 + x 2 2 + x 2 3 + x 2 4 + x 2 5 + 3x 2 6 + 3x 2 7 + 9x 2 8 , x 1 , x 2 , x 3 , x 4 , x 5 ≡ 0 (mod 3)}. Then by using the inclusion-exclusion principle we obtain N (1 6 , 3 2 ; n) = 15 (N (1 2 , 3 2 , 9 4 ; n) − 2N (1 1 , 3 2 , 9 5 ; n) + N (3 2 , 9 6 ; n)) …”

“…Therefore N (1 i , 3 j , 9 k ; n) cannot be determined for the 10 quadratic forms (i, j, k) = (7, 1, 0), (6, 1, 1), (5, 3, 0), (4, 3, 1), (3, 5, 0), (2, 5, 1), (1, 7, 0), (1,5,2), (1,3,4), (1,1,6) by using the method that we presented in this paper. Note also that the condition (iii) of Lemma 2.1 is not satisfied for these 10 quadratic forms.…”

“…, a 8 , we can assume that a 1 ≤ · · · ≤ a 8 . Under the simplifying assumptions (1.1) and (1.2), and with the conditions i + j + k = 8, j ≡ 0 (mod 2) and i, j, k ∈ N 0 , (1.3) there are 20 octonary quadratic forms (x 2 1 + · · · + x 2 i ) + 3(x 2 i+1 + · · · + x 2 i+j ) + 9(x 2 i+j+1 + · · · + x 2 i+j+k ), (1.4) where (i, j, k) = (8, 0, 0), (6,2,0), (4,4,0), (2,6, 0), (7, 0, 1), (6, 0, 2), (5,0,3), (4,0,4), (3,0,5), (2,0,6), (1,0,7), (3,4,1), (2,4,2), (1,4,3), (5,2,1), (4,2,2), (3,2,3), (2,2,4), (1,2,5), (1,6,<...>…”

On the Representations of a Positive Integer by Certain Classes of Quadratic Forms in Eight Variables

“…In our recent work [16], we constructed bases for the space of modular forms of weight 4 for the group Γ 0 (48) with character, and used modular forms techniques to determine the number of representations of a natural number n by certain octonary quadratic forms with coefficients 1, 2, 3, 4, 6. Finding formulas for the number of representations for octonary quadratic forms with coefficients 1, 2, 3 or 6 were considered by various authors using several methods (see for example [1,2,4,5,6,7]). In the present work, we adopt similar (modular forms) techniques to obtain the representation formulas.…”

Representations by octonary quadratic forms with coefficients 1, 3 or 9

“…, x 8 ) ∈ Z 8 | n = a 1 x with i + j + k + l = 8 under the conditions i ≡ j ≡ k ≡ l ≡ 0 (mod 2) or i ≡ j ≡ k ≡ l ≡ 1 (mod 2) appeared in literature. See [1], [2], [3], [4], [6], [9] and [11]. For convenience, we write (i, j, k, l) to denote an octonary quadratic form given by (1.1), and we write N(1 i , 2 j , 3 k , 6 l ; n) to denote the number of representations of n by the octonary quadratic form (i, j, k, l).…”